A three-span continuous beam ABCD with three equal spans supports a uniform load
ID: 1860260 • Letter: A
Question
A three-span continuous beam ABCD with three equal spans supports a uniform load of intensity q. Determine all reactions of this beam and draw the shear-force and bending-moment diagrams, labeling all critical ordinates.
Image can be found on this link since I can't upload onto chegg:
https://www.dropbox.com/s/6gjjc983mpnokrh/fig.png
Explanation / Answer
it is a statistically indeterminate problem
we need to solve it by superposition
case 1
consider only Reaction Ra and Rd with uniform distribution
case 2
consider onlt Ra,Rb,Rd and no uniform distribution
case 3
consider onlyRa, Rb, Rc, Rd nouniform distribution
case 1
we get y=(-q/24EI)*(x^4-2Lx^3+L^3x)
at x=L/3 y=-0.0113qL^4/EI
at x=2L/3 we get y=-0.113qL^4/EI
case 2)
when Rb is applied then deflection
with a=L/3 and b=2L/3
yb=-Pa^2b^2/3(EIL)=0.01646RbL^3/EI
case 3)
when Rc is applied
yc=0.01646RcL^3/EI
yb=(yb)w+(yb)Rb+(yb)Rc=>
-0.113qL^4/EI+0.01646RbL^3/Ei+0.01646RcL^3/RI=0
two more equation can be found out by force balance and moment balance
so reaction can be found hence bending moment dia can be found out