Introduction:
In case of beams supporting supporting uniformly uniformly distributed distributed load, the maximum maximum bending moment increas increases es with the square square of the span span and hence they become become unecono uneconomic mical al for long long span span structures. In such situations arches could be advantageously employed, as they would develop horizontal reactions, which in turn reduce the design bending moment.
Arch: An arch arch is an curved curved beam beam in which which horizo horizonta ntall moment moment at the the end is whol wholly ly (or) (or)
partially prevented .ence the horizontal trust will be introduced at the supports.
Type Type of arches:
!here are mainly three types of arches that tha t are commonly used in practice a. !hree hinged arch, ".!wo#hinged ".!wo#hinged arch and $.%ixed#fixed arch.
!hree#hinged arch is statically &eterminate structure
!wo#hinged !wo#hinged arch is statically indeterminate structure
%ixed#fixed arch are statically Indeterminate
structures
(c) Fixed- fixed arch
Theoretical arch: If an arch is to ta'e loads, say , *, and + and a vector diagram and
funicular polygon are plotted the funicular polygon is 'nown as the linear arch or theoretical arch.
Actual arch -
It is not possible to construct the arch in the shape of theoretical arch. o the
actual arch should be in the shape of parabolic, circular and elliptical.
Eddy’s theorem:
/ddy0s theorem states that 1!he bending moment at any section of an arch is proportional to the vertical intercept between the linear arch (or theoretical arch) and the centre line of the actual arch.2
Normal thrust and radial shear in an arch rib:
3et 4 be the inclination of the tangent at 5. If is the horizontal thrust and 6 the vertical shear at 5, the normal and radial components at the section 5 is given by, 7ormal
thrust,
7
8
cos4
9
:adial hear, : 8 6 cos4 # sin4
;roblems-
. A three#hinged parabolic arch of uniform cross section has a span of 3 and a rise of <$. It is sub=ected to uniformly distributed load of intensity >m as shown in %ig. how that the bending moment is zero at any cross section of the arch
6
sin4
%I?-
!otal load 8 l WL
6A96"
8
2
!a'ing moment of all the forces about hinge A ,
WL
6"3 #
2
2
8 @
!a'ing moment of forces left of hinge C about C , W L
A " 8
oment at any point 58@
2
8 H
*. A three#hinged parabolic arch of uniform cross section has a span of B@ m and a rise of @ m. It is sub=ected to uniformly distributed load of intensity @ '7>m as shown in %ig. *. how that the bending moment is zero at any cross section of the arch.
%ig-* !a'ing moment of all the forces about hinge A , yields : a 8 : b 8+@@C7 !a'ing moment of forces left of hinge C about C , one gets a b 8 DE@ C7 !he bending moment at any section x from the left end is x
5 8: ax# ay#@
2
2
!he equation of the three#hinged parabolic arch is
here
58@
Y =
( − x )
4 hx l
l
2
+ A F&3 of D'7>m covers left half span of +#hinged parabolic arch of span +Bm and central rise Gm. &etermine the horizontal thrust also find (i) " (ii) hear force (iii) 7ormal thrust (iv) :adial shear at the loaded quarter point. 'etch "&.
%ig + !a'ing moment of all the forces about hinge A
HA 8@ #6" x +B 9 D x G x 8@ 6" 8 GC7 ∴ 6A8ED'7 !a'ing moment of forces left of hinge C about C
c 8@ 96" G#" x G8@ "
[email protected]'7
[email protected] '7
here
( − x )
4h x l
y8
l
2
y8 (D x G x >+B*) (+B#) y8Bm " at 8 #
[email protected] x B 9 ED x #D x x D.E 8G '7.m
7ormal thrust 8 7 8 9
[email protected] $os *+.B 9 G $os BB.@D 8 DD.+* '7 8
[email protected] in *+.B J G in BB.@D 8 # @.@@
;arabolic arch at different levels of height-
L √ √ h 1
L √ √ h 2
38 √ h 1 + √ h 2
3*8 √ h 1 + √ h 2
orizontal trust for three hinged parabolic arch at d ifferent levels of heights-
W ( L−
8
a l1
)
h 1 L 2+ h 2 L 1
33*
D. %ind the horizontal trust and vertical reactions at the supports for a three hinged parabolic arch as shown in the figure.
L √ √ h 1
3 8 √ h 1 + √ h 2
8 Em
L √ √ h 2
3*8 √ h 1 + √ h 2
!a'ing moment of forces left of hinge C
6A8BDC7 6"8BC7
8 @m
D. A three hinged parabolic arch of span *@m and central rised Em carries a point load at Bm from left hand as shown in the figure D. %ind the end reactions and find the maximum bending moment.
FI!"#E - $
!a'ing moments about A 6A8ED C7,
6" 8 GC7
!a'ing moments about A 6A8ED C7,
6" 8 GC7
: A 8 GD.+ C7 , : " 8+D.BC7
!a'ing moments about c from left hand side A8 "8*@C7 aximum positive bending moment at &
&8++BC7# aximum negative bending moment 8 #E@ C7# 3/6/3# . !he number of independent equations to be satisfied for static equilibrium of a plane structure is A)
b) *
c)+
d)G
*. A three hinged arch is ########### structure. +. A three hinged parabolic arch having an udl of intensity w>m throughout the span Kl0 then the bending moment at any point is a)wl b) wl*>* c) zero d) wl>* D. In three hinged parabolic arch rise can be expressed as y8
( − x )
( − x )
4h x l
A)
y8
l
2
x ( l − x )
2h x l
b)
y8
l
2
c)
y8
l
2
d)
y8
( − x )
4h l
l
2
E. If the temperature increases, the rise of three hinged parabolic arch will ######### B. If the temperature increases, the horizontal trust of three hinged parabolic arch will ###### L. Increase in arch length due to temperature is expressed as ###### ..
A parabolic arch rib, *@ m span and + m rise is hinged at the abutments and the crown and carries a point load of @ '7 at L.E m from the left hand hinge. $alculate the horizontal thrust and the bending moment at a section L.E m form right hand hinge. hat is the value of the greatest bending moment in the arch, and where does it occurM
.* A parabolic arch hinged at the springings and crown has a span of *D m. !he central rise of the arch is Em. It is loaded with a uniformly distributed load of intensity *@ '7>m on the left Dm length. $alculate the direction and magnitude of reaction at the hinges .+. /xplain the theoretical arch and actual arch.
. . D. A three hinged parabolic arch of span Bm and rise D m is sub=ected to two point loads of @@ '7 and G@ '7 at the left and right quarter span points respectively. %ind the reactions at supports. %ind also the bending moment, radial shear and normal thrust at Bm from left support.
3/6/3# * . %ind the horizontal trust for the three hinged parabolic arch of length Kl0 and uniformly distributed throughout the span with an intensity w>m, having a central rise h at the center *. &ifference between two hinged arch and three hinged archM +. %or a three hinged parabolic arch find the normal trust and shear force by using the given data where 8 +E'7,68BB'7,and 48*@@ D. /xpress the 3, 3* values for the given diagram
. E. %ind the end reactions for the given diagram
B. %ind the horizontal reaction for the three hinged parabolic arch.
. A parabolic arch hinged at the ends and crown has a span of *@ m. !he central rise of the arch is D m. It is loaded with a uniformly distributed load of intensity * '7>m on the left G m length. $alculate (a) !he direction and magnitude of reaction at the hinges, (b) "ending moment, normal thrust and shear at Dm and Em from left end. *. A three#hinged parabolic arch hinged at the supports and at the crown has a span of *D m and a central rise of D m. It carries a concentrated load of E@ '7 at G m from left support and a uniformly distributed load of +@ '7>m over the left half portion. &etermine the moment, thrust and radial shear at a section B m from the left support. +. A parabolic arch, hinged at the ends has a span +@ m and rise E m. A concentrated load of * '7 acts at @ m from the left hinge. !he second moment of area varies as the secant of the slope of the rib axis. $alculate the horizontal thrust and the reactions at the hinges, D. A parabolic arch, hinged at the ends has a span +@ m and rise E m. A concentrated load of * '7 acts at @ m from the left hinge. !he second moment of area varies as the secant of the slope of the rib axis. $alculate the maximum bending moment anywhere on the arch.
