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

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

(OP)

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

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.

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.
## RE: How low yielding point in stress strain curve calculated in Explicit?

## RE: How low yielding point in stress strain curve calculated in Explicit?

Jeff

Pipe Stress Analysis Engineer

www.xceed-eng.com

## RE: How low yielding point in stress strain curve calculated in Explicit?

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:

## RE: How low yielding point in stress strain curve calculated in Explicit?

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

Jeff

Jeff

Pipe Stress Analysis Engineer

www.xceed-eng.com

## RE: How low yielding point in stress strain curve calculated in Explicit?

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....)

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