EDDY’S THEOREM ON ARCHES
EDDY’S THEOREM ON ARCHES
Actual Arch: The arch which follows either a parabolic, circular or elliptical shape and
is easily constructed with an aesthetic appearance called an actual arch.
Fig 1: Actual arch
Consider an arch (2 or 3 hinged) as shown in the figure subjected to the loads W1, W2
and W3. Let Va and Vb be the reactions at supports A and B. Let H is the horizontal
reaction at each support.
Linear or theoretical arch: The arch which follows a funicular polygon shape
after the application of a series of loads is called a linear or theoretical arch.
Fig 2: Linear arch
- Consider the funicular polygon – ACDEB of the arch as shown in the figure in which the members AC, CD, DE and EB are pin jointed and loaded with W1, W2 and W3 at points C, D and E.
- Generally, the members in the linear arch are subjected to compressive forces and joints must be in equilibrium.
Fig 3: Vector Diagram
- Referring to the vector diagram let PQ, QR and rs represent the loads W1, W2 and W3.
- Let OM represents Horizontal thrust, MP represents a vertical reaction at A and MS represents a vertical reaction at B of the arch.
- If the arch is provided as the same funicular shape (shown in fig 2 ) then the bending moment for such type of arch will be zero.
Fig 4: Combination of the linear arch and actual arch
The figure shows the combination of the actual arch and the linear arch. Let x be
the section to determine the bending moment, and y and y1 be the rises for actual
and linear arch respectively.
Bending moment at section X0-X = Hy
Bending moment at section X0-X1 = Hy1
The net bending moment at the overlapped portion of X section: H (y1 - y)
Therefore, net BM at section X is proportional to the difference in rise. i.e., (y1 - y)
Therefore Eddy’s Theorem states that "The bending moment at any section is proportional to the vertical intercept between the actual arch and the linear arch".
Er. SP.ASWINPALANIAPPAN., M.E.,(Strut/.,)
Structural Engineer
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