# Reactions of a redundant frame structure using Unit Load Method

• spggodd
In summary: Thanks for all your help by the way, it's much appreciatedIn summary, the conversation discusses a problem with a redundant structure and determining reactions at specific points using the unit load method. The individual has tried using equilibrium equations and removing supports at point D, but has encountered difficulties. They have been advised to calculate deflection at point D and use unit loads in the x and y directions. The individual has also used compatibility equations and expressions for bending moments to solve for the reactions at points A and D. They are now looking for a simpler method to check their answer.
spggodd

## Homework Statement

Hi, I am having trouble with the following problem and I can't seem to find any examples:

I am trying to determine reactions at points A and D of the redundant structure below using the unit load method, there is a 10kN load at the top left point (B), all lengths are 1m. Apologies for the poor diagram, please ask if you need clarification.

## Homework Equations

Equilibrium Equations?

## The Attempt at a Solution

I started by working out forces in the x direction.

10000+Dx+Ax =0

Then in the y direction:

Ay + Dy = 0

And then moments around D:

10000 x 1 + Ay

This then gives a series of un-solvable equations.
I think I may be ok when getting to the actual unit load method but I can't convince myself that I'm right for the equilibrium equations.

Can you help out?

Thanks
Steve

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spggodd said:

## Homework Statement

Hi, I am having trouble with the following problem and I can't seem to find any examples:

I am trying to determine reactions at points A and D of the redundant structure below using the unit load method, there is a 10kN load at the top left point (B), all lengths are 1m. Apologies for the poor diagram, please ask if you need clarification.

## Homework Equations

Equilibrium Equations?

## The Attempt at a Solution

I started by working out forces in the x direction.

10000+Dx+Ax =0
good
Then in the y direction:

Ay + Dy = 0
yes.
And then moments around D:

10000 x 1 + Ay
there is a fixed end moment A , M_Az, that you must include on this equation.
This then gives a series of un-solvable equations.
I think I may be ok when getting to the actual unit load method but I can't convince myself that I'm right for the equilibrium equations.

Can you help out?

Thanks
Steve
you have 3 equilibrium equations with 5 unknowns, so the frame is statically indeterminate to the second degree. You must remove the support at D, calculate the deflection of D in the x and y directions, and apply the unit loads at D in the x direction and then in the y direction. What have you tried beyond writing the equilibrium equations?

1 person
Ok I have got to the point where I have

Forces in X = 10000 - Ax - Dx = 0
Forces in Y = Ay + Dy = 0
Moments about D = 10000 x 1 - Ma + Ay = 0

Which gives me: Ax, Dx, Ay, Dy and Ma as my unknowns with 3 equations to solve.

-------------------------------

Removing supports at D and doing the same again gives:

Forces in X = 10000 - Ax = 0 -------> Ax = 10000 N
Forces in Y = Ay = 0 -------> Ay = 0 N
Moments about A = 10000 x 1 - Ma -------> Ma = 10000 Nm

Where next?
Is there any reference sources you can provide so I can have a look at any example problems?

I need to work out deflection in the x and y direction as you've mentioned above, both individually but how do you go about this. I assume you need to create 2 normal bending moment equations?

Last edited:
spggodd said:
Ok I have got to the point where I have

Forces in X = 10000 - Ax - Dx = 0
Forces in Y = Ay + Dy = 0
Moments about D = 10000 x 1 - Ma + Ay = 0

Which gives me: Ax, Dx, Ay, Dy and Ma as my unknowns with 3 equations to solve.
you need 5 equations, three from the static equilibrium equations and the other 2 from deflection compatability equations
-------------------------------

Removing supports at D and doing the same again gives:

Forces in X = 10000 - Ax = 0 -------> Ax = 10000 N
pointing left or right?
Forces in Y = Ay = 0 -------> Ay = 0 N
yes
Moments about A = 10000 x 1 - Ma -------> Ma = 10000 Nm
cw or ccw?
Where next?
Now calculate the deflection of joint D for that load case. You will need to know this, because ultimately, when you restore the support reactions at D, deflection is 0 at D.
Is there any reference sources you can provide so I can have a look at any example problems?
http://www.public.iastate.edu/~fanous/ce332/force/cdframefp.html
I need to work out deflection in the x and y direction as you've mentioned above, both individually but how do you go about this. I assume you need to create 2 normal bending moment equations?
Very tedious process, I am afraid. I'm glad I don't do these anymore. Yet, it is imperative that you understand the concept of how and why this works. Otherwise, you can shove this stuff into the computer and have no idea of the accuracy of the results. Very dangerous.

Hi PhantonJay, I think I may have got somewhere with this last night..

Replacing support D with two redundant reactions R1 and R2 gives the compatibility equations of:

ΔDy+a11R1+a12R2=0
ΔDx+a21R1+a22R2=0

Using the unit load method to calculate the displacements and flexibility coefficients using the equations:

ΔDy=∑∫L((M0M1y)/EI)dz

ΔDx=∑∫L((M0M1x)/EI)dz

a11=∑∫L((M1y2)/EI)dz

a22=∑∫L((M1x2)/EI)dz

a12=a21=∑∫L((M1yM1x)/EI)dz

The expressions for the bending moments of the frame are:

In DC:

M0=0
M1y=0
M1x=-1z1

In CB:

M0=0
M1y=-1z2
M1x=-1

In BA:

M0=10Z3
M1y=-1
M1x=-1(1-z3)

Substituting these values into the equations above and solving gives:

Δdy=1/EI(∫10(-10z3)dz3=-5/EI

Δdx1/EI(∫10(-(10z3)(1-z3))dz3=-1.666/EI

a11=1/EI(∫10(-z2)2)dz2+∫10((-1)2)dz3)=1.333/EI

a22=1/EI(∫10(-1z1)2dz1+∫10((-1)2)dz2+∫10(-1(1-z3))2dz3)=1.666/EI

a12=a21=1/EI(∫10(-1z2)(-1)dz2+∫10(-1)(-1(1-z3))dz3)=1/EI

Substituting these values into the compatibility equations gives:

-5/EI+(1.333/EI)R1+(1/EI)R2=0

And

-1.666/EI+(1/EI)R1+(1.666/EU)R2=0

Solving for R1 and R2 gives:

R1=5.459 kN
R2=-2.277 kN

-------------------------------

I'm convinced this is the right method however I am not overly confident with my expressions for the bending moments in each of the frame members?

Can I use another simpler method to check my answer etc?

spggodd: I currently think we cannot check your final answer until you post the E, I, and A value you used, where A = cross-sectional area of each member. Please post the E, I, and A values you assumed for your R1 and R2 answers.

Last edited:
Hi NVN,

To get rid of EI I just multiplied both sides by EI, this gets rid of the term right?

Also, where do I need to use A?Sent from my iPhone using Physics Forums

spggodd: I retract my statement in post 7 now. Excellent work. Your R1 and R2 answers in post 6 are correct. The correct answers are R1 = -2.273 kN, and R2 = 5.455 kN, where R1 is the horizontal reaction force at point D.

Last edited:
Ok, no worries.

Do you agree with the rest of my calculations though?Sent from my iPhone using Physics Forums

I'm glad I've finally seemed to crack this! Sent from my iPhone using Physics Forums

I've been going over this and I'm struggling to grasp how I got the bending moment equations for each member.

Also, my lecturer seems to specifically use a "System 0" and "System 1"... Etc.

The method in using seems to kind of match some of the examples, such that it uses the Stationary of Complementary Strain Energy equations and so forth.

I think I have the correct answers but I want to understand what I'm doing a bit better.

Is someone able to shed any light, or give a more detailed description of the process?

Thanks

## 1. What is the Unit Load Method and how does it differ from other methods of frame analysis?

The Unit Load Method is a simplified approach used for analyzing the reactions and internal forces in a redundant frame structure. It differs from other methods, such as the moment distribution method, by considering the entire structure as a series of individual beams with concentrated unit loads at key points. This allows for a more straightforward analysis of the frame's overall response.

## 2. How do you determine the reactions at the supports using the Unit Load Method?

To determine the reactions at the supports, the Unit Load Method requires the use of equilibrium equations and compatibility equations. The equilibrium equations involve taking moments about different points and setting them equal to zero. The compatibility equations involve analyzing the deflections of the structure under each unit load and ensuring that they are equal at points of overlap.

## 3. Can the Unit Load Method be applied to all types of frame structures?

Yes, the Unit Load Method can be applied to any type of frame structure, including both statically determinate and statically indeterminate frames. However, the method may become more complex for structures with a higher degree of indeterminacy.

The main advantage of the Unit Load Method is its simplicity and ease of use. It also provides a good approximation of the reactions and internal forces in a frame structure. However, the method may not be as accurate as other more complex methods, and it may not be suitable for analyzing highly irregular or asymmetric structures.

## 5. Are there any limitations to using the Unit Load Method?

Yes, there are some limitations to using the Unit Load Method. It is not suitable for analyzing structures with moving loads or dynamic loads, as it only considers the effects of static loads. Additionally, the method assumes that the frame members are linearly elastic, and it may not accurately predict the behavior of structures with significant non-linearities, such as plastic hinges.

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