Section forces when using prestressed reinforcement

[EricsonEncinaZ]

Hello,

I was trying to capture the internal forces of prestressed elements. For this purpose I created a 3D model with a simply supported beam with a point load in the middle of the span.

Beam model: displacement-based + fiber section + 20 elements + 2 Gauss-Lobatto IPs + Linear geometric transformation
Beam dimentions: b=0.2m h=0.6m L=10m
Material for beam section: elastic with Ec=30000MPa
Strand dimentions: 2 strands of 100mm2 each, located at -0.26m of the CG of the section
Material for PT strands: Steel02 with Es=200GPa, fyp=1500MPa, b = 0 and sigInit=1500MPa (it starts at max stress and always provides the same stress)
Point load at mid-span: 40 kN

To record the response of the first, mid and last element I used the following 3 commands:
- recorder Element -xml Element1.out -time -ele 1 section 1 force, for mid and last "-ele 10 20 section 2 force"
- [eleResponse 1 forces], for mid and last [eleResponse 10 forces] and [eleResponse 20 forces]
- [eleResponse 1 section 1 force], for mid and last [eleResponse 10 section 2 force] and [eleResponse 20 section 2 force]

Question 1: Despite the vertical deformation at mid-span is clearly affected by the prestressed strands, none of the methods to record/capture the internal forces shows the actions produced by the prestressed strands, neither at the section level nor at the element level. It is important to note that the values given by the 3 methods to record the forces just give the moment produced by the external point load = [No axial load, 100kN·m at mid-span and 0 at the supports], but the forces produced by the prestressed = [300kN axial load and contant moment -78kNm] are not shown. Why is this happening?

Question 2: The recorder command agrees with the [eleResponse $ele section $sec force] command, but the values from the "eleResponse $ele forces" are slighty different, as if the former commands were measuring at IPs that are not located at the ends of the elements, and since I specified a Lobatto integration scheme the 2 IPs of each element should be in the element ends. The "eleResponse $ele forces" command is giving the forces at both ends of the elements and are equal to the hand-calculated values considering just the external force. Are these methods effectively recording forces at IPs located in different positions?

The code I'm using is below and uses kN and m as its units:

wipe
logFile LogFile.txt

model BasicBuilder -ndm 3 -ndf 6

node 1 0.0 0 0
node 2 0.5 0 0
node 3 1.0 0 0
node 4 1.5 0 0
node 5 2.0 0 0
node 6 2.5 0 0
node 7 3.0 0 0
node 8 3.5 0 0
node 9 4.0 0 0
node 10 4.5 0 0
node 11 5.0 0 0
node 12 5.5 0 0
node 13 6.0 0 0
node 14 6.5 0 0
node 15 7.0 0 0
node 16 7.5 0 0
node 17 8.0 0 0
node 18 8.5 0 0
node 19 9.0 0 0
node 20 9.5 0 0
node 21 10.0 0 0

fix 1 1 1 1 1 0 0
fix 21 0 1 1 0 0 0

uniaxialMaterial Elastic 2 30000000.0
uniaxialMaterial Steel02 160 1500000.0 200000000. 0.0 15.0 0.925 0.15 0.0 1.0 0.0 1.0 1500000.

section Fiber 500 {
patch quad 2 20 60 -0.3 0.1 -0.3 -0.1 0.3 -0.1 0.3 0.1
layer straight 160 2 1e-4 -0.26 0.05 -0.26 -0.05
}

geomTransf Linear 3 0.0 0.0 1.0

element dispBeamColumn 1 1 2 2 500 3 -integration Lobatto
element dispBeamColumn 2 2 3 2 500 3 -integration Lobatto
element dispBeamColumn 3 3 4 2 500 3 -integration Lobatto
element dispBeamColumn 4 4 5 2 500 3 -integration Lobatto
element dispBeamColumn 5 5 6 2 500 3 -integration Lobatto
element dispBeamColumn 6 6 7 2 500 3 -integration Lobatto
element dispBeamColumn 7 7 8 2 500 3 -integration Lobatto
element dispBeamColumn 8 8 9 2 500 3 -integration Lobatto
element dispBeamColumn 9 9 10 2 500 3 -integration Lobatto
element dispBeamColumn 10 10 11 2 500 3 -integration Lobatto
element dispBeamColumn 11 11 12 2 500 3 -integration Lobatto
element dispBeamColumn 12 12 13 2 500 3 -integration Lobatto
element dispBeamColumn 13 13 14 2 500 3 -integration Lobatto
element dispBeamColumn 14 14 15 2 500 3 -integration Lobatto
element dispBeamColumn 15 15 16 2 500 3 -integration Lobatto
element dispBeamColumn 16 16 17 2 500 3 -integration Lobatto
element dispBeamColumn 17 17 18 2 500 3 -integration Lobatto
element dispBeamColumn 18 18 19 2 500 3 -integration Lobatto
element dispBeamColumn 19 19 20 2 500 3 -integration Lobatto
element dispBeamColumn 20 20 21 2 500 3 -integration Lobatto

timeSeries Linear 1 -factor 1.0
pattern Plain 1 1 {
load 11 0.0 -40. 0.0 0.0 0.0 0.0
}

initialize

recorder Element -xml Element1.out -time -ele 1 section 1 force
recorder Element -xml Element10.out -time -ele 10 section 2 force
recorder Element -xml Element20.out -time -ele 20 section 2 force

constraints Transformation
numberer RCM
system UmfPack
integrator LoadControl 1
test NormDispIncr 1e-10 20
algorithm KrylovNewton
analysis Static

analyze 1

puts "Disp mid-span node: [nodeDisp 11 2] m\n"

puts "\nMoment at ele 1 (left side) : [eleForce 1 6] kNm"
puts [eleResponse 1 forces]
puts [eleResponse 1 section 1 force]

puts "\nMoment at ele 10 (mid-span) : [eleForce 10 12] kNm"
puts [eleResponse 10 forces]
puts [eleResponse 10 section 2 force]

puts "\nMoment at ele 20 (right side): [eleForce 20 12] kNm"
puts [eleResponse 20 forces]
puts [eleResponse 20 section 2 force]

wipe

[selimgunay]

Why don't you apply the axial force and the moment as external loading?

[EricsonEncinaZ]

Thanks for your reply. The reason is because it is way easier to define 1 prestressed section and use that section lots of times than apply externals loads each time that section is used
Additonally the axial load, moment and deformations are influenced by the strain profiles developed at different sections along the beam when external loads or lateral building deformations are imposed, which is something I would like to capture.
Are the commands mentioned in the original questions actually missing the internally developed forces? or Am I making a mistake in my modelling?

[selimgunay]

When you apply the prestressing force as initial stress to the steel, that will not act as corresponding axial force and moment, because from static equilibrium the internal forces anywhere in your simply supported beam are computed by using the external loads and you don't have any additional external loading due to the prestress.

[EricsonEncinaZ]

Thank you for your reply.

So this means that the command "-ele $ele section $sec args" shows the external forces being applied in a specific section. At some point in the solution the integral of the stresses over the section should be performed, Is there any way to access those values so one can have the forces developed at the section? or the only option is to post-process based on section stresses?

About my second question on the original post, why does that difference occur? Should't they be the same?