The bridge structure is generally designed to be a continuous bridge, as it has the advantage of experiencing less bending moment and deformation compared to a simple beam.
In a continuous beam, in addition to the prestressing force (primary), secondary effects due to indeterminacy occur when prestress is applied
Let us consider a simple example of applying prestress to a two-span continuous beam with tendons arranged as in (a).
In this case, the prestressing will cause the bending moment (primary moment) shape as shown in (b)
The prestressing will also cause camber at point B.
The constraint at point B will produce a downward reaction force (Rb) against the camber, as shown in (c), which will cause an upward reaction force of Rb/2 at points A and C.
Due to this reaction force, the bending moment diagram as shown in Figure (d) is generated, which is called the Secondary Moment.
The final moment from the prestressing applied to the continuous beam is the sum of the primary and secondary moments, as shown in Figure (e).
The secondary moments generated by prestressing are generally in the opposite direction of the primary moments, which reduces the effect of prestressing.
Unlike considering the tension from prestressing (Primary) as an internal force when designing a member, the secondary moments are considered as external forces and applied as acting member forces.
(1) Structure
This example includes the calculation of primary, secondary and total prestressing moments of a two-span continuous beam.
Strand eccentricity from the centroid (ep): 400mm
Prestressing force applied (P) : 4,000kN
(1) Primary moment
The primary moment can be calculated by multiplying the prestress and eccentricity.
① Let’s calculate the tendon primary moments
② Bending moment diagram (B.M.D)
(2) Final prestressing moment calculation
Calculate the final moment using the reaction force caused by the primary and the secondary moments.
The final moment can be calculated simply by using the principle of least work.
① Basic structure
② Moment calculation (Section AB = Section CB, 0 ≤ x ≤ 20m)
③ Calculation of indeterminate forces
④ Free body diagram
⑤ Member force diagram
Shear Force Diagram (S.F.D)
Bending moment diagram (B.M.D)
(3) Secondary moment calculation
The secondary moment can be determined using the final and primary moments calculated earlier.
① Secondary moment calculation
② Bending moment diagram (B.M.D)
Using the midas civil program, the above example was structurally analyzed and the results are compared.
① Primary moment
② Secondary moment
③ Final moment
The results of the manual calculation and the structural analysis using midas civil are compared.
The comparison shows that the results are close to each other for both the cases
|
Comparison |
Moment (kN·m) |
|||
|---|---|---|---|---|
|
Primary moment |
Secondary moment |
Final moment |
||
|
Point A |
Manual calculation |
-1,600 |
0 |
-1,600 |
|
MIDAS CIVIL |
-1,600 |
0 |
-1,600 |
|
|
Point B |
Manual calculation |
-1,600 |
2,400 |
800 |
|
MIDAS CIVIL |
-1,600 |
2,396 |
796 |
|
In this content, we discussed the concepts of primary and secondary due to prestressing in PSC members and showed how to calculate them using a simple example.
Now, let's briefly summarize how to apply these values in design.
① Stress and displacement analysis
When reviewing stress, displacement, and other service loads, the member forces generated by prestressing are not considered separately but are included in the total member forces.
② Bending strength analysis
In the case of PSC members, after effective prestressing is applied, the neutral axis changes as the load increases in the member, and the strain in the strands changes as well.
In PSC members, the stresses in the strand at the ultimate limit state are calculated using the force equilibrium condition and strain compatibility, and the bending strength is determined using them.
In this process, the tension due to the primary has already been considered, so the primary is excluded from the load combination and only the secondary is considered.
In some cases, the P-M interaction diagram of the member with PSC is created, or in other unavoidable cases, the member force including the primary is applied and reviewed as an RC member.
However, this method is not recommended for accurate design because it does not accurately reflect the behaviour of the prestressing strand.
③ Shear strength analysis
In the case of shear strength, the compression force introduced to the member is considered, so the member force that considers both primary and secondary is used without separating them.
In this content, we discussed how to calculate the primary moment and secondary moment for the prestress continuous beam.
In the actual calculation, you will not have to calculate it directly, but I hope that this technical material will help you understand the concept of prestressing in continuous beams.
Primary moment
Secondary moment
PSC
Continous beam
Midas Civil