# Reaction at roller connecting two cantilever beams

## Homework Statement

The problem statement is as stated in the attached image, For a given E and cross section of both beams, I am to find the reaction at roller E that attaches beam AB and DC

## Homework Equations

There should be stress relation equations and reaction equations. First is to go through the different equations, and then after getting the reaction, the maximum stress at AB will depend directly on the reaction at E - reaction at D.

## The Attempt at a Solution

I was first thinking that I could treat this problem as two separate beams, and after getting the reaction forces at point B and point D, I can relate it with the deflection equation, saying

δe=δab+δdc . But then, how would I express the material properties of e?

I then thought of just listing out the forces -
then it will be Ra-2*5(the distributed force)+Rb+Rd-5+Rc=0, and next will be Re=Rd+Rb. But where will I gain the other 3 equations? Would it come from balancing the moment equations at A, E, and C?

Maybe a step in the right direction will help. thanks!

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PhanthomJay
Homework Helper
Gold Member

## Homework Statement

The problem statement is as stated in the attached image, For a given E and cross section of both beams, I am to find the reaction at roller E that attaches beam AB and DC

## Homework Equations

There should be stress relation equations and reaction equations. First is to go through the different equations, and then after getting the reaction, the maximum stress at AB will depend directly on the reaction at E - reaction at D.

## The Attempt at a Solution

I was first thinking that I could treat this problem as two separate beams, and after getting the reaction forces at point B and point D, I can relate it with the deflection equation, saying

δe=δab+δdc . But then, how would I express the material properties of e?

I then thought of just listing out the forces -
then it will be Ra-2*5(the distributed force)+Rb+Rd-5+Rc=0, and next will be Re=Rd+Rb. But where will I gain the other 3 equations? Would it come from balancing the moment equations at A, E, and C?

Maybe a step in the right direction will help. thanks!
Your compatability equation δe = δab + δdc is not correct. The deflection of joint e is the same as the deflection of δab and the same as the deflection of δdc, that is, δe = δab = δdc . Look at each beam separately: the beam ab has the applied load plus the unknown load E at the pin; the beam cd has its applied load and the unknown load E (opposite in direction) at the pin. Set the deflections of each beam equal to solve for E. Be sure to calculate the deflections of the cantilevers at the pin properly using superposition for each.

Jay says "Be sure to calculate the deflections of the cantilevers at the pin properly using superposition for each." Even though these beams are indeterminate propped cantilevers with spring supports at the free ends, formulas for cantilever deflection can be used to solve this.