Pier coping · ACI 318-25

A worked example — designing a pier-coping D-region

A pier coping is a classic D-region: beam theory does not apply, so it is designed with a strut-and-tie model. Here is the full workflow in AStrutTie — template → model → analysis → code checks — on a single-column pier coping.

Based on the AStrutTie 2020 Technical Book, Example 01 — the same case one of our verification fixtures is built from, run here under ACI 318-25.

Theory: Hammerhead pier caps — designing the cantilevers with strut-and-tie

Given

f_ck = 24 MPa (concrete)
f_y = 400 MPa (steel)
φ = 0.75 (strength-reduction factor)
Coping dimensions, bearing plates & service loads — entered in the template (see the drawing below)

①

The problem

A single-column pier coping carries the girder bearing reactions out to the column. Its overall geometry, bearings and loads define the design — typed straight onto the drawing.

Pier-coping geometry and dimensions — External geometry, bearings and loads of the pier coping.

②

Build the model — template → strut-and-tie

The pier-coping template builds a strut-and-tie model automatically: struts (compression, dashed), ties (tension, solid), and nodal zones typed CCC / CCT / CTT. It is then refined to follow the load path.

Strut-and-tie model of the coping in AStrutTie — The strut-and-tie model on the AStrutTie canvas — node, member and force labels.

③

Analyze & check — every member, every combination

Analysis returns the member force F_u in every strut and tie, then checks all of them against the code. Here: 33 / 33 strength checks satisfied, minimum safety factor 1.01 (governing tie m18), and all 3 development length checks pass.

Coping analysis results with member forces and STM verdict — Analysis · Check — STM verdict and member forces.

④

Tie check — required reinforcement

ACI 318-25

Required steel A_s,req is compared with the provided bars (size × count). φ = 0.75, f_y = 400 MPa.

A_s,req = F_u / (φ · f_y)

Required reinforcement table for the coping ties — Tie checks — required vs provided A_s per member.

⑤

Strut check — effective strength & width

ACI 318-25

Effective strength depends on strut type — Boundary (f_ce = 0.85 f_ck = 20.4 MPa) or Interior, reinforced (0.637 f_ck = 15.3 MPa). The required width must fit the available width.

f_ce = 0.85 · β_s · β_c · f_ck

w_req = F_u / (φ · f_ce · b) ≤ w_prov

Strut strength verification of the coping — Strut checks — f_ce, required width and safety factor per member.

⑥

Nodal-zone check — by node type

ACI 318-25

f_ce = 0.85 · β_n · β_c · f_ck

w_req = F_u / (φ · f_ce · b) ≤ w_prov

Node	β_n	f_ce (MPa)
CCC	1.0	20.4
CCT	0.8	16.3
CTT	0.6	12.2

Nodal-zone strength verification of the coping — Nodal-zone checks — each face against its node type.

⑦

Development length check — anchorage at the node

ACI 318-25 §23.8.2 · §25.4

The tie must be anchored at the node: the available anchorage ℓanc (§23.8.2, extended nodal zone) must reach the required development length ℓd (§25.4). Available = straight length along the tie — no hook-tail credit: ℓdh is measured to the outside of the hook, so the hook's contribution is already in the required length.

Tie anchorage verification of the coping — Development length check — available ℓanc vs required development ℓd.

These are the same checks AStrutTie runs — and Technical Book Example 01 is one of the cases our verification gate holds the engine to, unsafe-side zero. How we verify →

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