How to Use Eddy's Method to Analyze Masonry Domes

This tutorials provides instruction on using the online masonry dome analysis applet based on the graphical method of Henry T. Eddy.  This method, described in New Constructions in Graphical Statics by Henry T. Eddy (Eddy 1877), is similar to an analysis method for an arch.  The force polygons for each voussoir in the lune share a common pole, a, from which meridional forces and the outward horizontal thrust at the dome's base are calculated.  This method provides an equilibrium solution or curve assuming zero hoop forces in the dome; that is, the dome consists of a series of arches butt against each other.  For a more detailed background on this graphical analysis method, see the Methodology page.

Overview

This java applet was developed using Cabri Geometry II software by Texas Instruments.
To learn more about Cabri, visit www.cabri.com.

Three general rules to keep in mind when using this applet:

1. "Ctrl" key (Windows) + mouse drag will shift the viewing window.

2. In general, bright blue open-circle points are parameters that the user can control.

3. To align the dark blue equilibrium curve within the dome section, slide the pt "wt line" along the horizontal axis until point "j" aligns with point "i" on line "fO."

This tutorial will be divided into three parts as shown on the figure:

    I. User-Defined Parameters

    II. Representative Dome Section and Force Polygon

    III. Representative Lune in Plan.

I.  User-Defined Parameters

1. The four sliding scales located on the left of the screen enable the user to define the structure to be analyzed and how its force polygon will appear on the viewing screen.  The minimum and maximum of these values are shown and may not be modified.

   
   1. Dome Section Scale: Adjusts the appearance of the representative dome section in terms of 1 cm on the screen = n meters actual structure.
   2. Force Polygon Scale: Adjusts the force polygon in terms of of 1 cm on the screen = n kiloNewton's (kN) force.
   3. Material Unit Weight: Specifies the unit weight (kN/m³) of the masonry.  This will be treated as a uniform load on the dome.  Uniform live and snow loads and concentrated loads, etc., are considered as surcharge weight in this applet, and should not be included here.
   4. Lune Plan Hoop Forces: Adjusts the appearance of the hoop resultant forces on the plan view of the individual dome lune.

2. Six geometric parameters in blue text are located below the first two sliders.

   1. Exterior Radius: Slide the point "radExt." on the centerline of the representative dome section up or down to specify the exterior radius of the dome.  Note that the scales may require additional refinement as other parameters are defined.
   2. Thickness: Slide the point "radInt" on the centerline of the dome section.  This applet assumes if the user knows the interior radius of the dome, then the thickness is known, and vice versa.  Note that thickness is in units of centimeters.
   3. Angle of Embrace: The angle of embrace is fixed at 90 degrees, a hemispherical dome.
   4. Oculus Angle: Move the point "phi" on the dome section to modify the opening.  The oculus angle is the angle of the opening at the crown of the dome measured from the crown.  A dome with no oculus has a zero phi value.
   5. Theta: Move the point "Theta" on the lune plan view to modify this value.  Theta represents the number of lunes into which the dome is approximated in plan view.  The maximum theta is 15 degrees, which is equivalent to dividing the dome into 24 wedges.  Larger theta values reduce the accuracy of this analysis method.
   6. Angle fO: The angle of line fO with respect to the vertical axis is not critical.  However the point "fO" on the horizontal axis should be further left than the blue "wt line."  Line fO establishes the ratio at which the initial equilibrium curve, c, is elongated to fit within the dome section.

    

3. Using the geometric parameters, the applet returns the following parameters: the mean radius, thickness-to-radius (t/R) ratio, and the Vouss. Arc Length, u.

   1. Dead Load (kN): The dead load is calculated automatically by the geometry of the dome as defined by the user.
   2. Total Load (kN):  The total load represents the total load of the uniform and surcharge loads for the individual lune.  To find the total weight of the dome, multiply this value by (360/theta).
   3. Surcharge (kN): Surcharge values depend on the amount added by the user by dragging the surcharge points down (see Note II.C. below).
   4. Voussoir Width (m): The average width of each voussoir in plan view is calculated by the geometry of the dome and lune as defined by the user.
   5. Force Polygon Scale (kN/cm): The force polygon is forced to fit in the mean radius of the dome, ab.  The weight and applied loads and total load values on this lune are scaled to fit proportionally by the force polygon scale.

II.  Representative Dome Section and Force Polygon

1. The dome section is evenly divided into 16 voussoirs:

   1.    The u value represents the arc length of one voussoir along the mean radius of the dome, and is constant for all voussoirs.

   2.    The w value represents the mean width of each voussoir in plan, and increases from crown to base.

    

2. The median radius is indicated by the red arc in the section.

3. To add surcharge on a particular region of the dome, move the bright blue points downward.  Every other point is labeled with its voussoir number.  The points are located at the approximate center of gravity of each voussoir.  Note that this analysis assumes axis-symmetrical loading.  Surcharge applied at one location on this representative lune is assumed to occur in a ring around the dome.

4. For clarity purposes, several lines of the force polygons have been hidden.

5. Segment ab represents the total load on this lune, including self weight and applied loads.  Segment u(i-1), ui represents the gravity loads on voussoir i.

6. The red cubic curve ab connects the X-points for each voussoir's polygon in the case of existing tensile resistance in the dome.  For example, the triangle formed by au5x, where x is the fifth x-point from the top represents the equilibrium force polygon for the lune that includes the uppermost five voussoirs.  For a masonry dome with limited or no tension capability, this curve is not applicable.

7. Wt line: The blue weight line represents the force polygon for zero hoop force condition.  To adjust its location, slide the "Wt line" point along the horizontal axis to pinpoint its final location with respect to the pole a.

8. Equilibrium curve c: The equilibrium curve is the assumed line of thrust for this analysis method.  Its location can be adjusted by moving the weight line.

9. Line fO and pp: Move the point "fO" along the horizontal axis so it is left of the wt line.

10. Curve qq: This curve determines the elongation necessary of the equilibrium curve.  The user does not directly modify it.  Note that this curve should fit under line fO after the initial equilibrium curve is elongated.

11. Points j and i: Slide the weight line along the horizontal axis until point j on "wt line" is directly on point i on line "fO."  Equilibrium curve "c" should now fit within the dome section.

12. Move section: Move the point "move section" to shift the dome section around the screen.

13. Bottom of wall: Move the blue point "bottom of wall" to elongate the wall section, which acts only to enhance the perspective of the dome section.

III. Representative Lune in Plan

1. In the bottom left corner of the screen, internal Resultant Forces (kN/m) values are calculated for each voussoir i for the case in which tension forces are allowable.

   1.  Hoop values: to find the total hoop force of a voussoir, multiply this value by u, the arc length of one voussoir.

   1. Negative values indicate hoop forces are in tension.  Positive values indicate hoop forces are in compression.

   2. Note that hoop forces are maximum compression at the crown of the dome, decrease to near zero, and then become tensile.  This transition corresponds with the intersection of the hoop forces with the lateral faces of the lune in plan view (see figure below).

   2.  Meridional values: to find the total meridional force of a voussoir, multiply this value by w_i, the mean width of a voussoir.

   1. Meridional forces are positive and in compression.  Meridional forces increase from the dome's crown to base.

   2. Meridional force resultants listed are the values at the center of gravity of the voussoir.  From the force polygon, the meridional force resultant of voussoir i is given by the average length of the segments from point "a" to the i th x-point and the (i-1)th x-point, which define the voussoir's top and bottom faces.

   3.  Horizontal Thrust at Dome Base:  This value represents the outward horizontal thrust at the base of the lune.

2. Representative Lune in Plan

   1. Theta: Move point "theta" to define the number of individual lunes into which the dome is divided.

   2. Move lune plan:  Slide this point along the dome centerline to move the lune plan in the viewing window.

Return to applet.

Go to Methodology.

Return to Analysis of Masonry Domes HomePage.