Arch geometry determines all design characteristics. From the viewpoint of arch design, the outside and inside arcs, or surfaces of the roof, are segments of concentric circles separated by the thickness of the roof. The skewback slope cuts the arc, and its angular value equals half the included central angle of the arch. The rise of the arch measures the distance from the inner chord - equal to the span and cutting the inside arc to the center of the roof at midspan.
Hypothetical position of thurst in a simple sprung arch, when the bricks are in full contact at joints
.
2 H
-F
F
-H
-F
F T
h
r
2 S W
2
W 2
CENTER OF GRAVITY
LINE OF THRUST
CENTER OF PRESSURE
Ribbed arch roof
Ring arch roof
Bonded arch roof
Typical castable construction for sprung arch
When span, thickness and rise are established, all other dimensions can be calculated using the formulas in this section.
The stability of any arch will depend on its rise, thickness and weight, as well as the thermal properties of the
refractories. Hot strength and thermal expansion are particularly important.
Good arch design must take these factors into consideration.
Rise is normally expressed in inches per foot of span, or in terms of the central angle. They are directly related.
It should be easy to visualize a larger
for higher values. Silica roofs made with brick that maintain dimensional stability and hot strength almost to their melting point, normally rise from one to two inches per foot.
High-alumina refractories used in arch construction call for at least 1.608 inch per rise per foot of span. Basic refractories need 21/4 to three inches of rise in sprung arches. Insulating firebrick, which give up hot strength in exchange for low thermal conductivity, call for two to three inches of rise per foot of span.
For many applications, a 1.608 inch rise, about 15/8-inch, is a logical standard in that it meets normal requirements for strength and stability. The 60° central angle equals one sixth of a circle and the span equals the inside radius, so the number of brick required to build the roof is easy to calculate.
The reaction of refractory brick to furnace operation, i.e., heat-up, establishes the practical limit for roof rise. Operation and thermal expansion tend to push the brick upward, opening joints at the top and pinching brick at the bottom. Brick that soften at operating temperatures may become permanently deformed, shortening the radius of the arc and increasing the rise.
As the arch rises on heat-up, the line of thrust, the line of force along which the arch distributes the vertical and horizontal elements of its weight, shifts downward. As the line of thrust approaches a horizontal position in the arch, the horizontal force approaches its maximum value.
In some furnaces allowance for thermal expansion of the brick will limit upward movement of the arch. Steel casings can provide an allowance for expansion. Paperboard placed between brick will burn out and make room for expansion. In some cases, horizontal tie rods arc spring loaded or manually adjusted to permit thermal expansion of the refractories.
Without adequate provision for thermal expansion, the relationship between arch thickness and rise* of the cold arch must be such that the line of thrust does not drop out of the arch angle including a higher rise and a
shorter radius. On the other hand, the flatter roof with a smaller rise indicates a smaller included central angle and a longer radius.
Experiences suggests that a simple roof arch should rise not less than one nor more than three inches per foot of span. For any particular furnace, the rise selected should depend on operating conditions, chiefly temperature, thermal cycling and the refractories used.
Typically, stable fireclay arches rise from 11/2 to three inches per foot of span.
High temperatures and soaking heat call
Special Skew
Approx.
Line of Thrust
Ground Brick and Mortar Mixture
Header Course
Standard Skew
Approx.
Line of Thrust
Ground Brick and Mortar Mixture Header Course
Special skewback A recommened construction
Built-up skewback A recommened construction
Standard Skew
Approx.
Line of Thrust
Approx.
Line of Thrust
Buttress Angle
Top of Skewback
Bulk Insulation Buckstay
Buttress Block
Skewback No. 60-9
Built-up skewback A construction to be avoided.
A 60° skewback with buttress block.
Special skewback A construction occasionaly used
Arch with skewbacks supported outside of walls
Special Skew Special Shapes
Note: A stretcher course should never be used immediately below the skewback.
when it is heated. If it does, the arch will not be stable.
The line of thrust in a cold arch should lie within the middle third of the brick. Generally, selecting the proper combination of brick shapes and doing a professional job to assure face-to-face contact between brick will keep the thrust where it belongs.
In practical construction problems, the vertical and horizontal components are more important than the resultant force.
The walls, with or without steel supports, must carry the vertical force, and the horizontal binding, including buckstays and tie rods, must contain the horizontal force.
* Assuming that the absolute lower limit for the rise of the line of thrust is 1/4 inch (1/48) per foot of span, the rise (h) must exceed thickness (T) times the of the cosine of the central angle (0)plus 1/48 span (S). This implies that T should not exceed:
h-1/48S Cosine 0/2
For a more complete discussion of arch stresses see: J. Spotts Mcdowell, “Sprung-Arch Roots for High Temperature furnaces,” Blast Furnace and Steel Plant, September 1939.