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# Equilibrium check

## Equilibrium check

(OP)
Hi all! I am checking the equilibrium of forces along the applied load direction for my Abaqus model. The forces are not exactly in equilibrium. Please look at the picture and table attached. It is a Abaqus explicit analysis. Am I missing something while checking the equilibrium.
Replies continue below

### RE: Equilibrium check

1) Why would an explicit analysis not consider equilibrium ?

2) If it doesn't, what/how does it solve ?

3) NASTRAN has a table, a summary of the results, where it shows the constraints balance the applied loads. Does ABAQUS have similar ??
I wonder if you have something like "AUTOSPC" on, where the FE will add constraints as required for equilibrium ?

"Hoffen wir mal, dass alles gut geht !"
General Paulus, Nov 1942, outside Stalingrad after the launch of Operation Uranus.

### RE: Equilibrium check

odd, I git the opposite definition from another source (simscale.blog.com) ...

"Time-dependent vs. Time-independent Analysis
For all nonlinear and non-static analyses, incremental load (also known as displacement steps) are needed. In more simplistic terminology, this means we need to break down the physics/time relationship to solve a mathematical problem. To do this, we form two groups: either time-dependent or time-independent problems. To solve these problems, we commonly use ‘implicit’ and/or ‘explicit’ methods.

We refer to problems as ‘time-dependent’ when the effects of acceleration are pronounced and cannot be neglected. For example, in a drop test, the highest force occurs within the first few milliseconds as the item decelerates to a halt. In this case, the effect of such a deceleration must be accounted for.

In contrast, when loads are slowly applied onto a structure or surface (i.e., when a monitor is placed onto a table) the loading can be considered ‘quasi-static’ or ‘time-independent’. This is because the loading time is slow enough that the acceleration effects are negligible. For more time-dependent and time-independent examples, there are several projects in the SimScale Public Projects database. Some interesting examples are also depicted in Figure 01."

"Hoffen wir mal, dass alles gut geht !"
General Paulus, Nov 1942, outside Stalingrad after the launch of Operation Uranus.

### RE: Equilibrium check

Basically, Abaqus/Explicit can only solve for dynamic equilibrium where P-I (external minus internal forces) is not equal to 0 but to Ma (mass times acceleration - Newton’s second law). However, Abaqus/Explicit can be used for quasi-static analyses but there’s no sepcial setting for that since the analysis is always more or less dynamic. You just have to model your problem in such a way that dynamic forces become insignificant (kinetic energy is very small when compared with internal energy).

### RE: Equilibrium check

"To do this, we form two groups: either time-dependent or time-independent problems. To solve these problems, we commonly use ‘implicit’ and/or ‘explicit’ methods."

so these sentences are poorly written. I read to say "time-dependent" (ie dynamic) is implicit (and "time-independent" to be explicit.

should've been written as ...
"To do this, we form two groups: either time-dependent or time-independent problems. To solve these problems, we commonly use ‘explicit’ and/or ‘implicit’ methods."

"Hoffen wir mal, dass alles gut geht !"
General Paulus, Nov 1942, outside Stalingrad after the launch of Operation Uranus.

### RE: Equilibrium check

@rb1957 -

Here is a solver agnostic or a "numerical" description. Regardless of the commercial code you are using, these underlying numerical principles will be true.

Explicit numerical schemes assume equilibrium at t=0 and "solve" the system of equations by marching in time - nothing fancy. Typically, the time-step and duration is "small" so you can even get away with a single precision binary but if the duration is longer or the solver needs "lots of" time steps, then commercial code vendors will rightly recommend using a double precision binary because round-off errors start to pile up. Since the underlying reformulation is like solving a wave propagation problem, there is a strict constraint on the time step size due to the smallest (characteristic) element size. There is no concept of "convergence" in explicit schemes.

Implicit numerical schemes check for and enforce equilibrium. Implicit schemes solve the governing system of equations by "inverting" the stiffness matrix and incrementally iterating in time (or pseudo time in "static" analyses). In practice, no one inverts the stiffness matrix; clever numerical tricks are used to make the process efficient. Also, most implicit numerical schemes are "unconditionally stable". Convergence is a commonly experienced pain using implicit analyses.

If you are interested, look up the easiest numerical schemes out there - forward and backward Euler schemes. Alternatively, look up the documentation in any commercial code.

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### RE: Equilibrium check

ok, back to the original definition ... but can you read FEA's post and get the same meaning ?
Implicit = time independent (static, enforced equilibrium)
Explicit = time dependent (dynamic, body forces applied for equilibrium)

but then why isn't the OP's difference constant (or oscillating) ?
I'm assuming he's applying a sinusoidal input.

"Hoffen wir mal, dass alles gut geht !"
General Paulus, Nov 1942, outside Stalingrad after the launch of Operation Uranus.

### RE: Equilibrium check

I can guess but I am not certain about the source of oscillations in the results.

By the way, an implicit numerical scheme has nothing to do with time i.e., you can solve a time dependent problem with an implicit code. Static analysis is an engineering approximation in which "time" is treated as a "load carrier". An explicit scheme, due to its very nature, is evolving the solution in time.

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### RE: Equilibrium check

Could it be the stiffness of the structure ? If rigid then it'd respond like the input cycle, yes?, but the flexibility of the structure may make the dynamic response of the structure phase shift from the input load ?

"you can solve a time dependent problem with an implicit code" ... this is quasi-static, with body forces ignored or modelled (as input "loads").

"Hoffen wir mal, dass alles gut geht !"
General Paulus, Nov 1942, outside Stalingrad after the launch of Operation Uranus.

### RE: Equilibrium check

#### Quote (IceBreakerSours)

Implicit numerical schemes check for and enforce equilibrium. Implicit schemes solve the governing system of equations by "inverting" the stiffness matrix and incrementally iterating in time (or pseudo time in "static" analyses). In practice, no one inverts the stiffness matrix; clever numerical tricks are used to make the process efficient. Also, most implicit numerical schemes are "unconditionally stable". Convergence is a commonly experienced pain using implicit analyses.
Both explicit and implicit codes solve the same equation of motion. The difference is that explicit schemes use the previous time step, while implicit schemes use information from the the previous and the to-be-calculated time step, which increases the computational burden per time step.

Equilibrium is satisfied in the sense that the equation of motion is satisfied at the end of each time step (for non-linear dynamics, a time-step would also involve e.g., Newton-type iteration steps to solve the non-linear dependency of stiffness/contact/geometry and response) for both types of schemes.

### RE: Equilibrium check

#### Quote (sushil_k)

Hi all! I am checking the equilibrium of forces along the applied load direction for my Abaqus model. The forces are not exactly in equilibrium. Please look at the picture and table attached. It is a Abaqus explicit analysis. Am I missing something while checking the equilibrium.
The equation of motion, assuming stiffness and damping to be dependent on response, load being dependent on time and mass being constant:

F = ma --> F(t) = K(u)u + C(u)(du/dt) + m(d/dt^2(u))

Clearly, the external load is not equal to the "reaction force" K(u)u, as inertia and energy dissipation (due to e.g., viscous damping) are present in the equations. I suggest you review the theory manual properly before continuing this quite complicated analysis project of yours.

### RE: Equilibrium check

""you can solve a time dependent problem with an implicit code" ... this is quasi-static, with body forces ignored or modelled (as input "loads")."

In principle, you can solve any numerical solid mechanics problem with an implicit code - static, dynamic, viscoelastic, poroelastic, etc. Its practical considerations that sometimes force the choice in choosing an explicit scheme e.g., implicit schemes will run into convergence problems solving a crash because there is "a lot going on" in a short time span. Sometimes the code developers think it is a poor use of resources to invest in an implicit version of a method and leave the feature in explicit only because it is typically easier to implement. Some commercial codes have split their implicit and explicit capabilities to offer them as different products but that is a business decision, not a mathematical/numerical one.

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### RE: Equilibrium check

centondollar is exactly right - Whether a scheme is implicit or explicit doesn't matter; you are ultimately solving the same governing equations.

There are a few subtleties that don't concern modelers but are still good to be aware of - 1) Numerical schemes need not be implicit or explicit only; you can have variants. 2) Under the hood, implicit and explicit schemes do NOT solve the exact same mathematical formulation of the governing equations; the physical equations undergo a mathematical transformation so that the equations are conducive to be solved with either an explicit or an implicit numerical scheme (as is most often the case with commercial codes).

Also, there is a misconception out there about implicit in that one, somehow, gets more "accurate" results with implicit probably because the solution is converged. "Convergence" and accuracy have nothing to do with each other. Most commercial FEA engineering implicit and explicit codes try to be 2nd accurate but that is a practical consideration; peer-reviewed publications report 8th order or more accurate schemes.

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