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I am trying to learn how to work out the bending moment for progressively complex systems to model a beam. I am assuming only one axis of motion (transverse = $y$).

For the basic equations of motion, $σ$ represents stress and $ε$ represents strain in terms of longitudinal stretch of each segment of the beam resulting from transverse ($y$) displacement.

(1) Simple Spring

If you model the beam as a simple spring for which $E$ (Young's Modulus) is the spring constant and $I$ is the cross-sectional moment of inertia, according to this reference, you get an equation of motion and bending moment of:

$σ = Eε$

$M = EIy_{xx}$

According to Wikipedia though, it's the following, so I'm not sure about signs:

$M = -EIy_{xx}$

(2) Damped Spring

enter image description here

If you add a damper to the bar simulation, according to this reference, you get the following:

$σ= Eϵ-ηε_t$

$M=-EIy_{xx}-Iηy_{xxt}$

Question

I am having trouble understanding how to extrapolate these principles to increasingly complex beam models.

Three Element System:

For example, in a three element model you can have:

enter image description here

$σ + \frac{η}{E_2}\dot{σ} = E_1ε + \frac{η(E_1+E_2)}{E_2}\dot{ε}$

Four Element System:

Or in a four element model you might have:

enter image description here

$σ + (\frac{η_1}{E_1} + \frac{η_2}{E_2})\dot{σ} + \frac{η_1η_2}{E_1E_2}\ddot{σ} = (η_1+η_2)\dot{ε} + \frac{η_1η_2(E_1+E_2)}{E_1E_2}\ddot{ε}$

How would you similarly go about expressing the bending moment with these types of more complex models?

My Guess

From looking at the two known equations above, except for the signs which I don't know how to anticipate, it looks like the equation for the bending moment can be defined somewhat as follows.

Two Element System:

$σ= Eε-η\dot{ε}$

$M=-EIy_{xx}-Iηy_{xxt}$

$M = -I(σ_{bend})$ (which roughly matches this ref)

$σ_{bend}= Eε_{bend}-η\dot{ε}_{bend}$

$ϵ_{bend} = y_{xx}$

Three Element System:

If so then for the three element system we might have:

$σ_{bend} + \frac{η}{E_2}\dot{σ_{bend}} = E_1ε_{bend} + \frac{η(E_1+E_2)}{E_2}\dot{ε_{bend}}$

$M = E_1Iy_{xx} + I\frac{η(E_1+E_2)}{E_2}y_{xxt} - \frac{η}{E_2}\dot{M}$

Is this the right idea? What do you do with signs here? It seems like the signs don't match to the equations of motion in the two given examples so I'm not sure if there's something more to be done with them.

Thanks for any help.

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