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The science
Why prestress concrete
Concrete is strong in compression and weak in tension (it cracks at about fr = 0.62 √f′c MPa). A stretched steel tendon squeezes the girder before the load arrives, so the load first has to remove that compression before any tension can appear. Tension is positive in every readout and colour map here (blue = compression, white = zero, orange to red = tension up to the cracking stress).
Stress at a section
With force P at eccentricity e below the centroid and a load moment M: σbottom = −P/A + (M − P e) yb/I and σtop = −P/A − (M − P e) yt/I. The tendon moment P e(x) is the red curve in the moment graph; what is left (blue) is the net bending the concrete feels. The I section is built from three rectangles; its area, centroid and second moment are checked against strip integration.
Load balancing
A parabolic tendon with sag e pushes up on the girder like a uniform load wp = 8 P e / L2. When the dead load equals wp the girder has no bending at all and carries a uniform compression P/A (numerical check 3). The red arrows under the girder are wp, the arrows on top are dead plus live load.
Camber and deflection
The shape comes from numerically integrating the curvature y″ = Mnet(x) / (E I) twice with zero deflection at both bearings. Closed forms give a midspan sag 5 w L4/(384 E I) and a camber 5 P e L2/(48 E I) for a parabolic tendon (check 2). Creep multiplies the sustained part by 1 + φ(t) with φ(t) = φu (t − t0)/(t − t0 + 35), which is why a girder keeps rising in storage. After cracking, the live-load stiffness drops with a Branson-type effective inertia Ie = (Mcr/M)3 Ig + [1 − (Mcr/M)3] Icr with Icr = 0.3 Ig (a simplification).
Losses
Elastic shortening Ep/Eci fcgp Aps (half of that for sequentially stressed post-tensioned tendons), friction P(1 − e−(μα + kL/2)) with μ = 0.2 and k = 0.0015 per metre, a lumped anchorage draw-in of 6 mm, and long-term creep, shrinkage (400 microstrain) and relaxation (4 % of the jacking force) growing as t/(t + 35). Strength gain follows f′c(t) = f′c t/(4 + 0.85 t) (ACI 209). The deck slab is added at day 28 and the live load at day 56.
Capacity
The nominal moment comes from strain compatibility with a power-formula strand stress curve (1860 MPa strand) and a Whitney stress block, compared with the ACI approximate fps = fpu(1 − 0.28 ρp fpu/(β1 f′c)) (check 4). Demand is the factored 1.2 D + 1.6 L moment against 0.9 Mn.
Honest simplifications
Rectangular parts for the section properties, no deck composite action, one lumped friction and creep model rather than a step-by-step time analysis, a single effective prestress at midspan used along the girder, no transfer-length effects, no shear, no mild reinforcement, and an I-section with fixed flange sizes. Numbers are textbook-scale and not a bridge design.