# Equilibrium equations of elastic body

• Engineering
• Mechanics_student
Mechanics_student
Homework Statement
Equilibrium equations of elastic body in contact with an elastic body
Relevant Equations
balance of forces and moment

Consider a structure comprising an elastic deformable body (in pink) attached to a rigid body (in yellow) at the right side ( $\Gamma_c$ interface) and fixed at the left side.

Assume a force $F$ is applied to the rigid body at an angle $\beta$, as shown in the diagram. Subsequently, the deformable body undergoes deformation. (By the way, the elastic and rigid bodies stay together even when the force is removed)

What are the equilibrium conditions for the rigid body?

I think that it is expressed as follows:

where

$\sigma$ is the stress of the elastic body and $d$ is the perpendicular distance from the forces causing moment to the reference point which is taken to be the bottom point of the left corner of the rigid body.

Note:

• x-axis is taken as the horizontal axis and y-axis as the vertical axis.
• the cantilever is allowed to deform and to rotate while staying attached to the rigid body.
I'd like to know whether the equations I wrote are correct?

How complicated do you want to make this? Seeing ##\Omega## to describe a body brings nightmares from the past to me.

I'd assume beam equations if you're all right with a simple approach. That is, the pink body is treated as a 1D body represented by its neutral line along its length. That implies the forces and torques done by the pink body on the yellow body are at the point where they meet.
(They only meet at a point because it's a 1D body)

Notice I shaped the yellow body as an L to take into account the possible misalignment with the neutral line of the pink body. Take into account that the shape of the yellow body isn't relevant. I'd have drawn it as an inclined straight line from B to C and the net effect would be the same.
Finally, to find the reactions / internal forces at point B, you have to study that joint as a fixed link capable of transmitting forces and torques.

The forces and torques at B can be translated into the stress in the pink body at that point if you're interested in that. Knowing the reactions at A and B you can find the stress state of the pink body using beam equations.

On the other hand, if you're trying to solve it through a General 3D Elasticity approach the problem becomes way more complicated, and most likely there isn't an analytical solution to it. That's one of the reasons FEA is so widely spread.

By the way, try taking a look at the LaTex guide to use it correctly in the Forum.
https://www.physicsforums.com/help/latexhelp/

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Mechanics_student
Thank you so much for your useful reply! Ok, so, the problem I'm considering is solved in 2D.

Are the balance equations I have mentioned in my question correct?

Also, Is it possible to determine the stress ##\sigma## if we only know the magnitude of the force ##F## and the angle ##\beta##?
of course in addition to knowing the material properties and dimensions of the elastic body (beam).

Thank you.

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Mechanics_student said:
Thank you so much for your useful reply! Ok, so, the problem I'm considering is solved in 2D.
My worst fears are materializing before my eyes. Continuous mechanics is hard.

Mechanics_student said:
Are the balance equations I have mentioned in my question correct?
I don't think so.

Mechanics_student said:
Also, Is it possible to determine the stress ##\sigma## if we only know the magnitude of the force ##F## and the angle ##\beta##?
of course in addition to knowing the material properties and dimensions of the elastic body (beam).

Thank you.
If all boundary conditions are known it's doable even if only through numerical methods. Doing it by hand nowadays doesn't make much sense. There are specific software developed and user-friendly enough to help you with the task. Still, I think it's very useful you get the skills to at least write the necessary equations so you later know what you're doing when you're working with computers.
I'll try to dust what I remember from this topic and write another reply with how I'd face the problem.

In the meantime, I recommend you watch this video.

Juanda said:
My worst fears are materializing before my eyes. Continuous mechanics is hard.

I don't think so.

If all boundary conditions are known it's doable even if only through numerical methods. Doing it by hand nowadays doesn't make much sense. There are specific software developed and user-friendly enough to help you with the task. Still, I think it's very useful you get the skills to at least write the necessary equations so you later know what you're doing when you're working with computers.
I'll try to dust what I remember from this topic and write another reply with how I'd face the problem.
My actual problem is actually bigger than this and has more equations and conditions and my goal is to model my state problem and to solve it using a numerical software (FEM). But I needed to correct the equations of equilibrium of the rigid body and I simplified the problem further in order to only tackle this problem I posted here.

I am a little sad because my professor was telling me that its a standard knowledge and I was thinking to myself that it should very simple and I should know how to solve it. But from what I have realised that it's not getting enough attention and it seems many people like me are getting confused about it.

My professor said I need first to make sure the basic are correct before the numerical computation.

For example I got somehow a partial answer in this website and I'm not sure it helps me.

https://engineering.stackexchange.c...elastic-body?noredirect=1#comment116560_61401

Juanda said:
In the meantime, I recommend you watch this video.

Thanks! I'll watch it

Let me start by saying I don't think you're aware of how difficult is the problem you're proposing. I believe you drew something that looked simple as an attempt to better understand what you're studying but it's not simple at all.
You have the right attitude but my recommendation is that you stay fixed with the problems from your textbooks before trying to imagine the problems yourself.

In elasticity, there's the relation between forces and displacements. That is initially introduced as ##F = -kx##.
However, in general elasticity, that becomes significantly more complicated. The relation is still true in the way that displacements are proportional to forces but it's necessary to take a detour.
Picture source
I haven't read the link in detail myself because I was only interested in the picture.

How familiar are you with the diagram I showed?

Mechanics_student said:
My actual problem is actually bigger than this and has more equations and conditions and my goal is to model my state problem and to solve it using a numerical software (FEM). But I needed to correct the equations of equilibrium of the rigid body and I simplified the problem further in order to only tackle this problem I posted here.
What's your FEM software? With a computer, the problem becomes almost trivial. It's a matter of correctly defining the boundary conditions. The software will translate that into the necessary equations and solve them by itself.
Regarding the yellow rigid body, you just need to tell your software that the nodes attached to the yellow body will remain undeformed. In FEMAP-Nastran that's what's usually called an RBE2. Using a RBE2 it's possible to apply the force at a distant node so the possible introduction of torques will be correctly computed.

Mechanics_student said:
I am a little sad because my professor was telling me that its a standard knowledge and I was thinking to myself that it should very simple and I should know how to solve it. But from what I have realised that it's not getting enough attention and it seems many people like me are getting confused about it.

My professor said I need first to make sure the basic are correct before the numerical computation.
I don't think it's standard knowledge at all to be honest although I guess that depends a little on the context. Solving these problems by hand using generalized elasticity is not usual as far as I know.
Depending on the degree of accuracy you're required then it's doable. For example, the procedure I offered in post #2 should give you an approximation and, depending on how slender is the beam (and a few other considerations), the results will be fairly accurate.

Mechanics_student said:
For example I got somehow a partial answer in this website and I'm not sure it helps me.

https://engineering.stackexchange.c...elastic-body?noredirect=1#comment116560_61401
The answer you got there is basically the same one I offered in post #2. It's using beam theory to solve the problem but you said that's not enough for you.

Mechanics_student
For the overall force balance in the x-direction, I get $$F_{Ax}=F\cos{\beta}$$where ##F_{Ax}## is the x-direction reaction force component on the beam exerted by the cantilever support at A.

For the overall force balance in the y-direction, I get $$F_{Ay}=F\sin{\beta}$$where ##F_{Ay}## is the y-direction reaction force component on the beam exerted by the cantilever support at A.

For the overall moment balance centered at A, I get $$M_A=F\sin{\beta}L$$where ##M_A## is the counterclockwise moment at A exerted by the cantilever support at A on the beam, and L is the total length of the beam.

The moment balance for the section of the beam between A (x = 0) and arbitrary location x, I get $$M_A-xF\sin{\beta}=M(x)$$where M(x) is the clockwise moment exerted by the part of the beam beyond x on the part of the beam up to x, or $$M(x)=(L-x)F\sin{\beta}$$From the beam bending equation , we have $$EIy''=-M(x)=-(L-x)F\sin{\beta}$$subject to the boundary conditions ##y'=y=0## at x =0.

Juanda, Mechanics_student and Lnewqban
Mechanics_student said:
What are the equilibrium conditions for the rigid body?

I think that it is expressed as follows:

View attachment 350767
It seems that your diagram is missing important information.
Could you please define the n and d variables in your equations?

What is the exact point of application of the force on the yellow rigid body?
How are those equations applicable to the equilibrium conditions for the rigid (yellow) body?

Chestermiller said:
overall moment balance centered at A,
Chestermiller said:
For the overall force balance in the x-direction, I get $$F_{Ax}=F\cos{\beta}$$where ##F_{Ax}## is the x-direction reaction force component on the beam exerted by the cantilever support at A.

For the overall force balance in the y-direction, I get $$F_{Ay}=F\sin{\beta}$$where ##F_{Ay}## is the y-direction reaction force component on the beam exerted by the cantilever support at A.

For the overall moment balance centered at A, I get $$M_A=F\sin{\beta}L$$where ##M_A## is the counterclockwise moment at A exerted by the cantilever support at A on the beam, and L is the total length of the beam.

The moment balance for the section of the beam between A (x = 0) and arbitrary location x, I get $$M_A-xF\sin{\beta}=M(x)$$where M(x) is the clockwise moment exerted by the part of the beam beyond x on the part of the beam up to x, or $$M(x)=(L-x)F\sin{\beta}$$From the beam bending equation , we have $$EIy''=-M(x)=-(L-x)F\sin{\beta}$$subject to the boundary conditions ##y'=y=0## at x =0.
Hello, thank you! These equations are correct also in 2D right? I feel a bit confused regarding moving from 1D to 2D, because in 2D we'll have not only a point B connecting the two objects but a surface in common. In case I need to write the force in terms of stress of the elastic body, we divide by the area?

Juanda said:
Let me start by saying I don't think you're aware of how difficult is the problem you're proposing. I believe you drew something that looked simple as an attempt to better understand what you're studying but it's not simple at all.
You have the right attitude but my recommendation is that you stay fixed with the problems from your textbooks before trying to imagine the problems yourself.

In elasticity, there's the relation between forces and displacements. That is initially introduced as ##F = -kx##.
However, in general elasticity, that becomes significantly more complicated. The relation is still true in the way that displacements are proportional to forces but it's necessary to take a detour.
Picture source
I haven't read the link in detail myself because I was only interested in the picture.
View attachment 350800

How familiar are you with the diagram I showed?

What's your FEM software? With a computer, the problem becomes almost trivial. It's a matter of correctly defining the boundary conditions. The software will translate that into the necessary equations and solve them by itself.
Regarding the yellow rigid body, you just need to tell your software that the nodes attached to the yellow body will remain undeformed. In FEMAP-Nastran that's what's usually called an RBE2. Using a RBE2 it's possible to apply the force at a distant node so the possible introduction of torques will be correctly computed.
View attachment 350804

I don't think it's standard knowledge at all to be honest although I guess that depends a little on the context. Solving these problems by hand using generalized elasticity is not usual as far as I know.
Depending on the degree of accuracy you're required then it's doable. For example, the procedure I offered in post #2 should give you an approximation and, depending on how slender is the beam (and a few other considerations), the results will be fairly accurate.

The answer you got there is basically the same one I offered in post #2. It's using beam theory to solve the problem but you said that's not enough for you.
The basic thing is that I needed to write the full equilibrium equations of the rigid body as my professor said. But then I realized that I might need to substitute for the stress which is done by using Euler bernoulli assuming small displacement, but then I was hesitant again of whether I just keep the stress general, so.

Mechanics_student said:
The basic thing is that I needed to write the full equilibrium equations of the rigid body as my professor said. But then I realized that I might need to substitute for the stress which is done by using Euler bernoulli assuming small displacement, but then I was hesitant again of whether I just keep the stress general, so.
Equilibrium of the yellow body can be found as previously stated.
Juanda said:
Finally, to find the reactions / internal forces at point B, you have to study that joint as a fixed link capable of transmitting forces and torques.
View attachment 350777

The forces and torques at B can be translated into the stress in the pink body at that point if you're interested in that. Knowing the reactions at A and B you can find the stress state of the pink body using beam equations.

For that, you're missing some key information as mentioned by @Lnewqban.
Lnewqban said:
What is the exact point of application of the force on the yellow rigid body?

The application of beam theory from @Chestermiller at #9 is correct too assuming that the horizontal component of the input force ##F## is aligned with the neutral line of the pink beam so it doesn't introduce an additional torque on the system.
(There is the topic of introduced torques once deformed but that's a whole different story)

I think all the questions you (@Mechanics_student) had are now clear. Solving this problem from a General Elasticity point of view is way more complex and doesn't seem to be the objective of the exercise.

Lnewqban
Lnewqban said:
It seems that your diagram is missing important information.
Could you please define the n and d variables in your equations?

What is the exact point of application of the force on the yellow rigid body?
How are those equations applicable to the equilibrium conditions for the rigid (yellow) body?
Ok so, the original problem is that the elastic body is piezoelectric. And no only that, it is a bimorph. It is about optimization but taking into account the applied force. It is a big problem but my professor was only commenting on the equations regarding the rigid body (equilibrium) so he asked me to correct them.

Juanda said:
Equilibrium of the yellow body can be found as previously stated.

For that, you're missing some key information as mentioned by @Lnewqban.

The application of beam theory from @Chestermiller at #9 is correct too assuming that the horizontal component of the input force ##F## is aligned with the neutral line of the pink beam so it doesn't introduce an additional torque on the system.
(There is the topic of introduced torques once deformed but that's a whole different story)

I think all the questions you (@Mechanics_student) had are now clear. Solving this problem from a General Elasticity point of view is way more complex and doesn't seem to be the objective of the exercise.
Thanks. So as stated in answer #9, these equations are also for 2D? What about if I wanted to write the equations not only interms of forces only but also in terms of the stress of the elastic body.

Mechanics_student said:
Hello, thank you! These equations are correct also in 2D right? I feel a bit confused regarding moving from 1D to 2D, because in 2D we'll have not only a point B connecting the two objects but a surface in common. In case I need to write the force in terms of stress of the elastic body, we divide by the area?
The development I presented uses the so-called strength of materials approach, which involves making an educated guess with regard to the kinematics of the deformation. This leads to the final equation that I presented in my post. A full two dimensional theory of elasticity development is much more complicated.

Chestermiller said:
The development I presented uses the so-called strength of materials approach, which involves making an educated guess with regard to the kinematics of the deformation. This leads to the final equation that I presented in my post. A full two dimensional theory of elasticity development is much more complicated.
Thank you. May I know please why it becomes much complicated in 2D?

Do these equations make sense for example?

$$\sum F_x = 0 \quad \Rightarrow \quad F_x - \int_{\Gamma_c} \sigma_x \, dA = 0$$
$$\sum F_y = 0 \quad \Rightarrow \quad F_y - \int_{\Gamma_c} \sigma_y \, dA = 0$$
$$\sum M = 0 \quad \Rightarrow \quad F_y \cdot d_x - F_x \cdot d_y - \int_{\Gamma_c} (\sigma_x \cdot y - \sigma_y \cdot x) \, dA = 0$$

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Mechanics_student said:
.... It is a big problem but my professor was only commenting on the equations regarding the rigid body (equilibrium) so he asked me to correct them.
So, has your professor found your equations shown in post #1 above to be incorrect?
What subject is he testing?

Mechanics_student
Mechanics_student said:
Thank you. May I know please why it becomes much complicated in 2D?
Because continuous mechanics is a pretty hard topic.
Juanda said:
In elasticity, there's the relation between forces and displacements. That is initially introduced as ##F = -kx##.
However, in general elasticity, that becomes significantly more complicated. The relation is still true in the way that displacements are proportional to forces but it's necessary to take a detour.
Picture source
I haven't read the link in detail myself because I was only interested in the picture.
View attachment 350800

How familiar are you with the diagram I showed?

Mechanics_student said:
Do these equations make sense for example?

$$\sum F_x = 0 \quad \Rightarrow \quad F_x - \int_{\Gamma_c} \sigma_x \, dA = 0$$
$$\sum F_y = 0 \quad \Rightarrow \quad F_y - \int_{\Gamma_c} \sigma_y \, dA = 0$$
$$\sum M = 0 \quad \Rightarrow \quad F_y \cdot d_x - F_x \cdot d_y - \int_{\Gamma_c} (\sigma_x \cdot y - \sigma_y \cdot x) \, dA = 0$$

If I were you I'd try the approach given in previous posts using beam theory / strength of materials.
In most cases, the results are accurate enough.
Beam theory is already capable of going from the loads on the line to the stress fields on the body.

The equations you have by themselves don't contain enough information to be solved because you don't know the pressure distribution between the pink and yellow bodies.

Physically describing reality can be as complex as you want. It's best to start with a simple model and see if that's enough for your task. Taking into account that you seem to be struggling, I don't think you're ready to try to tackle this from a general elasticity point of view unless you're using modeling software that simplifies the task for you.
Juanda said:
What's your FEM software? With a computer, the problem becomes almost trivial. It's a matter of correctly defining the boundary conditions. The software will translate that into the necessary equations and solve them by itself.
Regarding the yellow rigid body, you just need to tell your software that the nodes attached to the yellow body will remain undeformed. In FEMAP-Nastran that's what's usually called an RBE2. Using a RBE2 it's possible to apply the force at a distant node so the possible introduction of torques will be correctly computed.
View attachment 350804

Last edited:
Mechanics_student and Lnewqban
The 2D equations should include:$$\frac{\partial \sigma_{xx}}{\partial x}+\frac{\partial \sigma_{xy}}{\partial y}=0$$$$\frac{\partial \sigma_{xy}}{\partial x}+\frac{\partial \sigma_{yy}}{\partial y}=0$$The above are the stress-equilibrium equations.

u = v = 0 at x=0

##\boldsymbol{\sigma}=0## on the free surface

##\sigma_{xx}A=-F\cos{\beta}## at x = L

##\sigma_{xy}A=-F\sin{\beta}## at x = L

plus the strain displacement equations and Hooke's Law in 2D

Mechanics_student
Lnewqban said:
So, has your professor found your equations shown in post #1 above to be incorrect?
What subject is he testing?
I'll ask him and let you know soon, as he's away for a conference.

Juanda said:
Because continuous mechanics is a pretty hard topic.

If I were you I'd try the approach given in previous posts using beam theory / strength of materials.
In most cases, the results are accurate enough.
Beam theory is already capable of going from the loads on the line to the stress fields on the body.

The equations you have by themselves don't contain enough information to be solved because you don't know the pressure distribution between the pink and yellow bodies.

Physically describing reality can be as complex as you want. It's best to start with a simple model and see if that's enough for your task. Taking into account that you seem to be struggling, I don't think you're ready to try to tackle this from a general elasticity point of view unless you're using modeling software that simplifies the task for you.
It's Sfepy

Chestermiller said:
The 2D equations should include:$$\frac{\partial \sigma_{xx}}{\partial x}+\frac{\partial \sigma_{xy}}{\partial y}=0$$$$\frac{\partial \sigma_{xy}}{\partial x}+\frac{\partial \sigma_{yy}}{\partial y}=0$$The above are the stress-equilibrium equations.

u = v = 0 at x=0

##\boldsymbol{\sigma}=0## on the free surface

##\sigma_{xx}A=-F\cos{\beta}## at x = L

##\sigma_{xy}A=-F\sin{\beta}## at x = L

plus the strain displacement equations and Hooke's Law in 2D
Thank you so much!

Mechanics_student said:
It's Sfepy
Not the kind of software I had in mind for sure.
I guess there really are Python libraries for everything.
I'll probably try to solve it myself too using the equations and boundary conditions provided by @Chestermiller at #20.
Chestermiller said:
The 2D equations should include:$$\frac{\partial \sigma_{xx}}{\partial x}+\frac{\partial \sigma_{xy}}{\partial y}=0$$$$\frac{\partial \sigma_{xy}}{\partial x}+\frac{\partial \sigma_{yy}}{\partial y}=0$$The above are the stress-equilibrium equations.

u = v = 0 at x=0

##\boldsymbol{\sigma}=0## on the free surface

##\sigma_{xx}A=-F\cos{\beta}## at x = L

##\sigma_{xy}A=-F\sin{\beta}## at x = L

plus the strain displacement equations and Hooke's Law in 2D

However, wouldn't there be an additional boundary condition forcing the end of the pink beam to keep its shape because it's fixed to a rigid body?
Or maybe not consider it fully fixed if it's just pushing it but at least its rotation should be linked. I'm trying to say it allows for expansion due to the Poison ratio, but the line should remain straight even if it becomes inclined.

Last edited:
Mechanics_student
Juanda said:
Not the kind of software I had in mind for sure.
I guess there really are Python libraries for everything.
I'll probably try to solve it myself too using the equations and boundary conditions provided by @Chestermiller at #20.

However, wouldn't there be an additional boundary condition forcing the end of the pink beam to keep its shape because it's fixed to a rigid body?
Yes. The equations for the stresses at L should involve integrals over the area rather than stress times area, and there should be a kinematic condition requiring the ratio of the displacements to be constant. ##u=kv##

Mechanics_student
Juanda said:
Not the kind of software I had in mind for sure.
I guess there really are Python libraries for everything.
I'll probably try to solve it myself too using the equations and boundary conditions provided by @Chestermiller at #20.

However, wouldn't there be an additional boundary condition forcing the end of the pink beam to keep its shape because it's fixed to a rigid body?
Or maybe not consider it fully fixed if it's just pushing it but at least its rotation should be linked. I'm trying to say it allows for expansion due to the Poison ratio, but the line should remain straight even if it becomes inclined.
Thank you! The displacement should be zero at the fixed end (at the left) because of the fixed support and also no rotation is allowed at the fixed support

Chestermiller said:
Yes. The equations for the stresses at L should involve integrals over the area rather than stress times area, and there should be a kinematic condition requiring the ratio of the displacements to be constant. ##u=kv##
##u## is the displacement along x-axis and ##v## is the displacement along y-axis?

Last edited:
Mechanics_student said:
Thank you! The displacement should be zero at the fixed end (at the left) because of the fixed support and also no rotation is allowed at the fixed support
That's true. I was talking about the rightmost end though. See post #25 for more details. You can see how ##u=kv## corresponds to the equation of a line because that end will remain straight since it's fixed to the straight rigid body. If the rigid body turns (and it should turn according to the fixation at A and the input forces at L), the pink body will turn with it similarly if we consider them fixed.

Mechanics_student said:
##u## is the displacement along x-axis and ##v## is the displacement along y-axis?
Yes. That's how it's usually denoted. And ##w## for the ##z##-axis in case you want to consider deformations/displacements due to the Poison ratio.

Last edited:
Mechanics_student
Mechanics_student said:
##u## is the displacement along x-axis and ##v## is the displacement along y-axis?
Yes

Mechanics_student
Chestermiller said:
Yes. The equations for the stresses at L should involve integrals over the area rather than stress times area, and there should be a kinematic condition requiring the ratio of the displacements to be constant. ##u=kv##
Do I also perform inner product of stress with the normal vector of the surface when I integrate the stress over the area?

Mechanics_student said:
Do I also perform inner product of stress with the normal vector of the surface when I integrate the stress over the area?
Of course

Mechanics_student
Chestermiller said:
Of course

If I name the area as ##\Gamma_c## then it could be written this way ?

Chestermiller

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