Explanation for Wolfe's Method of Graphical Analysis

The methodology for this method of graphical analysis for domes is based on William S. Wolfe's 1921 text Graphical Analysis: A text book on graphic statics.  It is plausible he based his text on methods previously published by other engineers and architects, such as J. W. Schwedler and Henry T. Eddy.

Based on the membrane theory of domes, Wolfe's method assumes internal forces act along an imaginary membrane with zero thickness at the median radius of the dome section.  Similar to the analytical approach to the membrane theory, this graphical method provides only an approximation of internal forces in domes.  This method approximates the dome as a number of slices or lunes in which two opposing lunes form a quasi two-dimensional arch (Heyman 1977).

After the user defines the geometric parameters of the dome, such as radius, thickness, and angle of embrace, the volume of each voussoir is determined by multiplying u, the arc length of each voussoir, by w, the mean width of each voussoir in plan, and the thickness t; each value can be determine by a simple geometric relation.  This volume is multiplied by the material unit weight, giving the dead load of each voussoir, and added to the surcharge weight.  The sum determines the vertical length between points i and (i + 1) on the vertical segment of the force polygon (see Figure 1).

The horizontal line at the top of the force polygon represents the thrust line at the crown of the dome at its centerline where gravity loads are theoretical zero.  Each voussoir has five forces acting on it: its self weight and applied loads, two meridional forces on its top and bottom faces, and two lateral hoop forces from adjacent voussoirs.

Meridional forces act in a longitudinal direction of a dome and increase from crown to base in magnitude.  Meridional forces are due to the weight of the masonry and applied loads.  Voussoir 3, for example, experiences meridional forces from voussoirs 1 and 2 above it, and also voussoirs 4 through 16 below.  On the force polygon, the meridional forces on the voussoir's top and bottom faces are represented by the diagonal dashed vectors labeled "Meridional force 2" and "Meridional force 3," respectively.  To estimate the average meridional force within the voussoir, the mean of these values is taken.  The meridional force is then converted into a stress resultant with units of force per length.  The length of the voussoir on which the meridional force acts is the mean voussoir width, w₃.

Hoop forces act in a latitudinal direction of a dome on the lateral faces of the voussoirs, and contribute significantly to the stability of domes and their construction.  For a dome with uniform axisymmetric loading, hoop forces from adjacent slices are equal in magnitude and act on the same plane.  On the representative lune plan view, the hoop forces act perpendicular to lune's lateral faces.  The horizontal components of the hoop forces on each side of a voussoir are equal and opposite.  The resultant of their forces produces a net force toward or away from the dome center, as shown on the force polygon. The magnitude of the net force is the distance between intersections of consecutive meridional lines on the horizontal ray.

The magnitude of the hoop force is the length of the segment drawn perpendicular to the lune edge truncated by the complementary line. This complement represents the hoop force on the opposite lateral face of the voussoir.  These lines originate from the intersection of the meridional forces with the horizontal ray.  As a result, the magnitude of the hoop forces depends on the meridional force values in this method of analysis.

On the representative lune plan view, the hoop forces shown represent hoop force resultants with units of force per length.  To find the magnitude of the hoop force, multiply the magnitude listed under the hoop force resultants by u, the arc length of each voussoir.

On the force polygon, for domes with angles of embrace exceeding about 52 degrees, the meridional force vectors become steeper toward the base of the dome.  The hoop force lines intersect below rather than above the horizontal ray.  The transition represents the transition from compressive to tensile hoop forces in the dome.

Limit state analysis assumes that masonry has limited or no ability to resist tension forces.  Tension in the dome causes adjacent lunes to move apart and the base of the dome to splay.  For the case of no allowable tension, the hoop forces below the horizontal ray are assumed to equal zero.  Subsequently, a revised thrust line is drawn with segments connecting the vertical weight line on the force polygon to point "masonry" on the horizontal ray. To modify the thrust line, the user must slide this point to the location of the last compressive hoop force.  This relocates the line of thrust away from the median radius of the dome.  If the modified line of thrust remains within the thickness of the dome, then the dome is stable.  Otherwise, it may be necessary to increase the thickness at the base of the dome.

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