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**Arches**

**Three Hinged Arches**

- 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_{A}x - wx^{2}/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. - Three Hinged Semicircular Arch of Radius R carrying a UDL over the whole span.H = wR/2

M_{x}= -wR^{2}/2 [sin θ - sin^{2}θ]M_{max}= -wR^{2}/8

BM_{c}= 0

Point of contraflexure = 0 - 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** - Three Hinged Semicircular Arch Carrying Concentrated Load W at CrownH = V
_{A}= V_{B}= W/2

**Temperature Effect on Three Hinged Arches**

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^{0}C- H α 1/h

Where, H = horizontal thrust

and h = rise of arch - % 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.

- Two hinged semicircular arch of radius R carrying a concentrated load 'w' at the town.

H = w/π

Two Hinged Circular arch - 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 sin
^{2}α - A two hinged semicircular arch of radius R carrying a UDL w per unit length over the whole span.Two Hinged Semicircular arch
- 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 - 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 archH = wl
^{2}/8h - When half of the parabolic arch is loaded by UDL, then the horizontal reaction at support is given byTwo Hinged parabolic arch
- When two hinged parabolic arch carries varying UDL, from zero to w the horizontal thrust is given byTwo Hinged parabolic archH = wl
^{2}/16h - A two hinged parabolic arch of span l and rise h carries a concentrated load w at the crown.

H = 25 wl/ 128 h

Two Hinged parabolic arch

where H = Horizontal thrust for two hinged semicircular arch due to rise in temperature by T^{0}C.

where l0 = Moment of inertia of the arch at crown.

H = Horizontal thrust for two hinged parabolic arch due to rise in temperature T^{0}C.

**Reaction Locus for a Two Hinged Arch**

**Two Hinged Semicircular Arch**

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

**Eddy's Theorem**

M_{x}αy

where, M_{X} = BM at any section

y = distance between given arch linear arch

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