How low yielding point in stress strain curve calculated in Explicit? 2

[Mechanicslearner]

Hello,

I am experimenting with non linear cantilever beam with axial point load at tip where stress levels decrease after reaching plasticity. You can see as follows

Yield stress------------- plastic strain
210000000------------- 0
232300000------------- 0.00884475
255000000------------- 0.01858908
278100000------------- 0.02823539
260000000------------- 0.03798338
241500000------------- 0.04764082
212000000------------- 0.05725989
246100000------------- 0.06648743
291600000------------- 0.07557343
327000000------------- 0.08462176
363000000------------- 0.0935831
399600000------------- 0.10245897

I know that if we use true stress and true strain , stress wont decrease that much but I am doing this just to learn how explicit works
in abaqus, I tried in implicit and abaqus standard where it gives error when stress levels try to decrease. Explicit worked great, where it gives little fluctuation curve when I apply very small load for each time step and vice versa (large fluctuations when load applied in small time).

So the question is how explicit calculates stress fluctuations

TIME IN
SECONDS------------- FORCE IN NEWTONS------------- STRESS IN N/M^2------------- PLASTIC STRAIN------------- MAX DISPLCAEMENT
0.------------- 0. ------------- 0. ------------- 0.------------- 0.
200.007E-03------------- 6.80023E+06------------- 13.622E+06 ------------- 0.------------- 648.472E-06
400.002E-03 ------------- 13.6001E+06 ------------- 27.1997E+06------------- 0.------------- 1.29395E-03
600.007E-03 ------------- 20.4002E+06------------- 40.8579E+06 ------------- 0.------------- 1.94486E-03
800.003E-03 ------------- 27.2001E+06------------- 54.3996E+06 ------------- 0.------------- 2.5888E-03
1.00001------------- 34.0002E+06------------- 68.07E+06 ------------- 0.------------- 3.24075E-03
1.2 ------------- 40.8001E+06------------- 81.5996E+06 ------------- 0.------------- 3.88445E-03
1.40001 ------------- 47.6002E+06 ------------- 95.261E+06 ------------- 0.------------- 4.5362E-03
1.6 ------------- 54.4001E+06------------- 108.8E+06 ------------- 0.------------- 5.18074E-03
1.80001 ------------- 61.2003E+06 ------------- 122.44E+06 ------------- 0.------------- 5.83151E-03
2. ------------- 68.0001E+06------------- 136.E+06 ------------- 0.------------- 6.47742E-03
2.2 ------------- 74.8001E+06------------- 149.625E+06 ------------- 0.------------- 7.127E-03
2.4 ------------- 81.6002E+06------------- 163.238E+06 ------------- 0.------------- 7.77555E-03
2.60001 ------------- 88.4002E+06------------- 176.803E+06 ------------- 0.------------- 8.42251E-03
2.80001 ------------- 95.2002E+06 ------------- 190.4E+06 ------------- 0.------------- 9.06965E-03
3.00001 ------------- 102.E+06 ------------- 204.E+06 ------------- 0.------------- 9.71818E-03
3.20001 ------------- 103.943E+06 ------------- 207.97E+06 ------------- 0.------------- 9.90617E-03
3.40001------------- 105.886E+06 ------------- 211.768E+06 ------------- 0.0007012------------- 16.4519E-03
3.6 ------------- 107.829E+06 ------------- 215.663E+06 ------------- 0.002246 ------------- 32.7624E-03
3.8 ------------- 109.771E+06 ------------- 219.774E+06 ------------- 0.003877 ------------- 48.9672E-03
4. ------------- 111.714E+06 ------------- 223.426E+06 ------------- 0.005325 ------------- 64.0021E-03
4.2 ------------- 113.657E+06 ------------- 227.311E+06------------- 0.006866 ------------- 79.2692E-03
4.40001 ------------- 115.6E+06 ------------- 231.197E+06------------- 0.008422 ------------- 95.2867E-03
4.6 ------------- 117.543E+06 ------------- 235.213E+06 ------------- 0.01012 ------------- 112.795E-03
4.8 ------------- 119.486E+06 ------------- 239.044E+06 ------------- 0.01175------------- 129.487E-03
5.00001 ------------- 121.429E+06 ------------- 242.854E+06 ------------- 0.01338------------- 145.941E-03
5.2 ------------- 123.371E+06 ------------- 246.741E+06------------- 0.01504------------- 163.372E-03
5.4 ------------- 125.314E+06------------- 250.732E+06 ------------- 0.01675 ------------- 180.792E-03
5.60001 ------------- 127.257E+06------------- 254.512E+06------------- 0.01835 ------------- 197.395E-03
5.8 ------------- 129.2E+06 ------------- 258.397E+06 ------------- 0.02000------------- 214.264E-03
6. ------------- 131.143E+06 ------------- 262.302E+06 ------------- 0.02164------------- 231.331E-03
6.20001 ------------- 133.086E+06------------- 266.199E+06 ------------- 0.02327------------- 248.149E-03
6.40001 ------------- 135.029E+06 ------------- 270.054E+06 ------------- 0.02495 ------------- 265.467E-03
6.60001 ------------- 136.971E+06 ------------- 274.E+06 ------------- 0.02700 ------------- 286.744E-03
6.8 ------------- 138.914E+06------------- 318.392E+06 ------------- 0.08671------------- 613.639E-03
7. ------------- 140.857E+06 ------------- 282.372E+06 ------------- 0.1006------------- 989.896E-03
7.20001 ------------- 142.8E+06 ------------- 341.619E+06 ------------- 0.1006------------- 992.474E-03
7.4 ------------- 144.743E+06------------- 289.048E+06 ------------- 0.1006 ------------- 989.228E-03
7.6 ------------- 146.686E+06 ------------- 300.987E+06 ------------- 0.1006 ------------- 991.31E-03
7.80001 ------------- 148.629E+06------------- 328.723E+06 ------------- 0.1006------------- 992.341E-03
8.00001------------- 150.572E+06 ------------- 300.798E+06 ------------- 0.1006------------- 989.983E-03
8.2 ------------- 152.514E+06------------- 304.804E+06 ------------- 0.1006------------- 990.702E-03
8.40001 ------------- 154.457E+06------------- 313.286E+06 ------------- 0.1006------------- 992.008E-03
8.6 ------------- 156.4E+06 ------------- 342.808E+06 ------------- 0.1006------------- 993.072E-03
8.80001 ------------- 158.343E+06 ------------- 349.E+06 ------------- 0.1006------------- 993.362E-03
9. ------------- 160.286E+06 ------------- 329.6E+06 ------------- 0.1006------------- 993.183E-03
9.20001 ------------- 162.229E+06------------- 324.321E+06------------- 0.1006------------- 993.244E-03
9.4 ------------- 164.171E+06 ------------- 328.262E+06 ------------- 0.1006------------- 994.491E-03
9.60001 ------------- 166.114E+06 ------------- 332.198E+06 ------------- 0.1006------------- 996.326E-03
9.8 ------------- 168.057E+06 ------------- 336.127E+06 ------------- 0.1006 ------------- 998.583E-03
10. ------------- 170.E+06 ------------- 340.005E+06 ------------- 0.1006 ------------- 1.00211


as you can see stress peaks at 6.8 second and decrease after that. May I know how abaqus explicit reached that value? I heard explicit does not do iteration and it calculates later time from current step. Unknown values are obtained from information already known.

 

[Mustaine3]

Explicit uses an very small time increment, so it can be used for highly dynamic analysis. Maybe you have dynamics in your structure that you don't see with an output at every 0.2s and at 6.8s you have an min or max value in your movement.

 

[JGard1985]

As mentioned, explicit is time depedent. In your example you need to be careful that when you apply your force, you acheive a static, steady state solution. If you slam your force (time less than natural period) you do not set up a natural frequency vibration in your subject. If you want to see how explicit compares to implicit for a steady-state load, I suggest you increase material damping to damp out the vibrations, and ensure you let the analysis run for a minimum of 3 times the natural period of the structure.

Jeff
Pipe Stress Analysis Engineer

http://www.xceed-eng.com

 

[Mechanicslearner]

Hello,

Thank you very much for the reply. Do you guys know how delta t is calculated in Abaqus explicit during velocity and displacement calculation? I read documentation in abaqus analysis user's guide, there they mentioned delta t as stability increment which is not substituted during numerical analysis.

Please read a short example of my analysis, which I am trying to learn:

Cantilever beam with axial load at the end. load = 1+ E9 N , total time = 60 seconds, step = explicit and linear (Nlgeom = OFF), fixed increment = 0.0001, fixed at other end, elastic modulus = 210 GPA , density 8050 kg/m^3 , poisson's ratio = 0.33

After analysis, the output are as follows, shown only first 4 increments:

 

[JGard1985]

Read this forum, timesteps in explict analysis need to be small enough to capture loads and deformation through a single element

http://www.eng-tips.com/viewthread.cfm?qid=311561

Jeff

Jeff
Pipe Stress Analysis Engineer

http://www.xceed-eng.com

 

[Mechanicslearner]

Hello JGard1985,

Thank you very much for the link. I already read that link before. 5th comment from bottom where henki says ///" BUT when I calculated the "I" in transverse direction, the time step size decreased to 8E-8s. This seems to determine the time step size, even though this latter stiffness has no effect on the results. I ran the beam analysis again using a fixed time increment of 1E-6s and the results were same as those obtained with 5E-8s. So this answers my question."///

Does this mean I have to calculate transverse moment of inertia instead second moment of inertia for natural frequency? if so what is formula for rectangular cross section as I could not find it, am sorry to say that.

Please read the short explanation of how I got the delta t(i+1) and delta t(i) values by using increment 1 and increment 2 values from above table:
Increment 1:
acceleration = 0.0828164369 m/sec^2, velocity = 0.00000414085616 m/sec, displacement = 0 (dont know why its zero)
according to Abaqus Theory guide 2.4.5 (

http://129.97.46.200:2080/v6.14/books/stm/default.htm)

velocity(i+0.5) = velocity(i-0.5) + [((delta t(i+1) + delta t(i))/2) * acceleration(i)]
so substituting increment 1 value gives
0.00000414085616 = 0 +[((delta t(i+1) + delta t(i))/2) * 0.0828164369]
(delta t(i+1) + delta t(i))/2) = 0.0000500004

Increment 2:
acceleration = 0.164693177 m/sec^2, velocity = 0.0000165164402 m/sec, displacement = 8.28178137*E-10
by using above method
(delta t(i+1) + delta t(i))/2) = 0.0000751433

but in this we have displacement value which makes it possible to get delta t(i+1) value from following equation
displacement(i+1) = displacement(i) + delta t(i+1)* velocity(i+0.5)
8.28178137*E-10 = 0 + delta t(i+1) * 0.0000165164402
delta t (i+1) = 0.0000501427
so delta t (i) = 0.0001001439

I know its amateur method of finding it but I did it to make sure what could be the values because Explicit does not do iterations so correct me if I am wrong in doing above calculation.

Now according to documentations:
delta t = L/c where c = sqrt(E/rho)
delta t = 2/omega , where omega is natural frequency
delta t = 2/omega * (sqrt(1+ ξ^2) - ξ) , where ξ is fraction of critical damping where I used default linear and quadratic bulk viscosity 0.06 and 1.2 respectively

from Roark's formula
omega = 1.732/2*pi * (sqrt(E*I*g/Wl^3)) because am trying to learn cantilever with axial load at tip with density = 8050 , poissons ratio = 0.33 , elasticity = 210 GPa , I know values are approximate for learning
I get closer to delta t values only by using roark's formuals, if there is any other formula for calculating natural frequency of cantilever beam let me know

I know this is a long reply but I typed to make it clear and I hope its easy for you guys to find delta t values. So please let me know if there is a correct natural frequency formula or alternative way for that and working on this for a week now