EDDY’S THEOREM ON ARCHES

EDDY’S THEOREM ON ARCHES

Actual Arch: The arch which follows either parabolic, circular or elliptical shape and are easily constructed with aesthetic appearance is called as actual arch.

Fig 1: Actual arch

Consider an arch (2 or 3 hinged) as shown in figure subjected to the loads W₁, W₂ and W₃. Let V_a and V_b are the reactions at supports A and B. Let H is the horizontal reaction at each support.

Linear or theoretical arch: The arch which follows funicular polygon shape after application of series of loads are called as linear or theoretical arch. 

Fig 2: Linear arch

•  Consider the funicular polygon – ACDEB of arch as shown in 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 is subjected to compressive forces and joints must be in equilibrium.

Fig 3 : Vector Diagram

• Referring to the vector diagram let pq,qr and rs represents the loads W₁, W₂ and W_3.
• Let OM represents Horizontal thrust, MP represents vertical reaction at A and MS represents 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 linear arch and actual arch

Figure shows the combination of actual arch and linear arch. Let x be the section to determine the bending moment, 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

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)

Therefor 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".