A three-span continuous beam ABCD with three equal spans supports a uniform load

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

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.

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

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