Tension stiffening via P-Delta analysis (CSI 1-016)

• elasticBeamColumn + geomTransf PDelta (Tcl engine only)
• Stage 1: gravity-combo axial tension buildup (nonlinear)
• Stage 2: target load case at end-of-Stage-1 tangent stiffness
• Same code path as production TclBuilder P-Delta
• Deflection & moment vs exact tension-stiffened beam (Timoshenko)

A slender pin-roller tie-rod carries a small transverse uniform load together with a large axial tension pulling the roller end. Tension measurably stiffens the beam against the transverse load — the deflection and moment both shrink relative to the no-tension case — the classic P-Delta tension-stiffening problem with a closed-form solution (Timoshenko 1956). STRIX solves this with its production P-Delta 2-stage static path (the Tcl-only route analysisHandler forces P-Delta submissions through): Stage 1 applies the axial tension in a nonlinear gravity combination to build up the member axial force under a geomTransf PDelta transformation; Stage 2 then solves the transverse load case at the tangent stiffness left at the end of Stage 1. CSI SAP2000 verification example 1-016 documents exactly this two-stage nonlinear-static method as its recommended way to solve tension-stiffening problems — it is the only method DCR implements, so this benchmark exercises the production code path directly, not an alternative formulation.

1. Uz (no tension)-5*w*L^4/(384*E*I)= −1.041667 in
2. My (no tension)w*L^2/8= 22.5 kip-in
3. Uz (w/ tension)Timoshenko eq.43: (5wL^4/384EI) * [1/cosh(u) - 1 + u^2/2] / [(5/24)u^4], u=sqrt(PL^2/4EI)= −0.543305 in
4. My (w/ tension)Timoshenko eq.45: (wL^2/8) * 2(cosh(u)-1)/(u^2*cosh(u))= 11.498079 kip-in

CSI SAP2000 Software Verification Example 1-016 'Tension Stiffening Using P-Delta Analysis'; independent reference = Timoshenko, Strength of Materials Part II, 1956, eq. 23 p.28 and eqs. 43/45 p.43. Reproduced from scratch by pd1Theory() to u=1.5, deflWithIn=-0.5433047259, momentWithKin=11.4980793, matching CSI's published -0.54330 in / 11.498 k-in exactly (exact unit conversion of the same physical problem, dimensionless u is unit-invariant).

The no-tension case uses the exact-for-prismatic-Euler-beam elasticBeamColumn element and is essentially mesh-independent (converged at any element count). The tension-stiffened case is genuinely mesh-sensitive: OpenSees’s PDelta geometric transform is a first-order (chord-rotation) approximation of the true hyperbolic tension-stiffened shape, so the discrete answer converges to the closed form as elements are added — a clean, roughly quadratic convergence is observed. At the finest mesh STRIX reproduces the exact Timoshenko tension-stiffened deflection and moment to a few hundredths of a percent, confirming both that the P-Delta pipeline converges to the right physics and that it correctly reduces to the plain linear-static answer when the axial tension is absent.

1. Timoshenko, S. — Strength of Materials, Part II. Eq. 23 p.28 (tension parameter u) and eqs. 43/45 p.43 (tension-stiffened deflection and moment).
2. CSI — SAP2000 Software Verification, Example 1-016: Tension Stiffening Using P-Delta Analysis (independent reference, Timoshenko 1956). CSI recommends the nonlinear-static P-Delta method used here.
3. OpenSees — geomTransf PDelta (P-Delta geometric transformation); elasticBeamColumn (Euler-Bernoulli); STRIX production TclBuilder Stage-1/Stage-2 P-Delta static path (Tcl engine only).

Engine

v1.0.6 (tcl)

Run date

2026-07-11

Record

records/PD1.json

Evidence archive

verif-evidence-eng1.0.2-win-x64.zip · PD1/

sha256

(pending publish)