Analysis of Arches & Cables | Structural Analysis - Civil Engineering (CE) PDF Download

Arches

Three Hinged Arches

1. Three Hinged Parabolic Arch of Span L and rise 'h' carrying a UDL over the whole span
   D_s = 0
   BM_c = 0
   H = wl2/8h
   M_x = V_Ax - wx²/2 - Hy where, H = Horizontal thrust
   V_A = Vertical reaction at A = wl/2
   Simply supported beam moment i.e., moment caused by vertical reactions.
   Hy = H-moment
   D_S = Degree of static indeterminacy
   BM_C = Bending Moment at C.
2. Three Hinged Semicircular Arch of Radius R carrying a UDL over the whole span.H = wR/2
   M_x = -wR²/2 [sin θ - sin² θ]M_max = -wR²/8
   BM_c = 0
   Point of contraflexure = 0
3. Three Hinged Parabolic Arch Having Abutments at Different Levels
   (i) When it is subjected to UDL over whole span
   (ii) When it is subjected to concentrated load W at crown
   
4. Three Hinged Semicircular Arch Carrying Concentrated Load W at CrownH = V_A = V_B = W/2

Temperature Effect on Three Hinged Arches

1. 

   Where, Δh = free rise in crown height
   l = length of arch
   h = rise of arch
   α = coefficient of thermal expansion
   T = rise in temperature in ⁰C
2. H α 1/h
   Where, H = horizontal thrust
   and h = rise of arch
3. % Decrease in horizontal thrust = δh/h x 100

Two Hinged Arches

Two hinged arch of any shape

D_S = 1
Where, M = Simply support Beam moment caused by vertical force.

1. Two hinged semicircular arch of radius R carrying a concentrated load 'w' at the town.
   H = w/π
   Two Hinged Circular arch
2. Two hinged semicircular arch of radius R carrying a load w at a section, the radius vector corresponding to which makes an angle α with the horizontal.Two Hinged Circular archH = w/π sin² α
3. A two hinged semicircular arch of radius R carrying a UDL w per unit length over the whole span.
   
   Two Hinged Semicircular arch
4. A two hinged semicircular arch of radius R carrying a distributed load uniformly varying from zero at the left end to w per unit run at the right end.
   Two Hinged Semicircular archH = 2/3.wR/π
5. A two hinged parabolic arch carries a UDL of w per unit run on entire span. If the span off the arch is L and its rise is h.
   
   Two Hinged parabolic arch
   H = wl²/8h
6. When half of the parabolic arch is loaded by UDL, then the horizontal reaction at support is given by
   
   Two Hinged parabolic arch
7. When two hinged parabolic arch carries varying UDL, from zero to w the horizontal thrust is given by
   
   Two Hinged parabolic arch
   H = wl²/16h
8. A two hinged parabolic arch of span l and rise h carries a concentrated load w at the crown.
   H = 25 wl/ 128 h

1. 
   where H = Horizontal thrust for two hinged semicircular arch due to rise in temperature by T ⁰C.
2. 
   where l0 = Moment of inertia of the arch at crown.
   H = Horizontal thrust for two hinged parabolic arch due to rise in temperature T ⁰C.

Reaction Locus for a Two Hinged Arch

1. Two Hinged Semicircular Arch
   Reaction locus is straight line parallel to the line joining abutments and height at πR/2
   
2. Two Hinged Parabolic Arch
   

Eddy's Theorem

M_xαy

where, M_X = BM at any section
y = distance between given arch linear arch