ASCE 7-22 Eq. 13.3-2 and 13.3-3 impose a lower and upper bound on the component seismic design force Fp. The bounds catch engineers in two ways: they make small, low-amplification components more demanding than the equation suggests (Fp,min), and they cap the demand for tall, flexible components on upper floors (Fp,max). Knowing when each governs is the difference between a clean submittal and a back-and-forth with the plan reviewer.

The three forces every submittal must compute

From ASCE 7-22 §13.3:

  • Fp = 0.4 · SDS · Ip · Wp · (Hf/Rμ) · (CAR/Rpo) — Eq. 13.3-1.
  • Fp,max = 1.6 · SDS · Ip · Wp — Eq. 13.3-2.
  • Fp,min = 0.3 · SDS · Ip · Wp — Eq. 13.3-3.

The design force is min(Fp,max, max(Fp, Fp,min)).

When Fp,min governs

The minimum bound is 0.3 · SDS · Ip · Wp. Compare to Fp = 0.4 · SDS · Ip · Wp · (Hf/Rμ) · (CAR/Rpo). The minimum is reached when (Hf/Rμ) · (CAR/Rpo) ≤ 0.75.

That happens for:

  • Rigid, low-elevation components in ductile buildings — e.g., a transformer at the ground floor of a building with Rμ = 1.5 and CAR = 1.0, Rpo = 1.5 → factor = 1.0/1.5 · 1.0/1.5 = 0.44 → Fp,min governs.
  • Most floor-mounted MEP equipment on lower levels of essential facilities.
  • Architectural components below grade or in mechanical rooms.

Engineers expect Fp to scale with CAR and Hf; they forget that Fp,min is a hard floor. Result: an under-designed anchor that fails plan review at the minimum-bound check.

When Fp,max governs

The maximum bound is 1.6 · SDS · Ip · Wp. Fp,max caps the demand at 4× the minimum, regardless of what the equation produces. It governs when (Hf/Rμ) · (CAR/Rpo) ≥ 4.0.

That happens for:

  • Tall flexible components (large CAR) on upper floors of stiff buildings (large Hf, low Rμ).
  • Cooling towers, elevated tanks, and stacks on roofs.
  • Long roof-mounted antenna or louvre systems.

Without Fp,max, the equation would predict design forces beyond the response that the building can deliver to the component, because the equation does not cap floor acceleration at the building's peak. The cap brings the design back to a defensible envelope.

Worked example — small chiller, ground floor, hospital

  • SDS = 1.5 g, Ip = 1.5, Wp = 4,000 lb.
  • z/h = 0 → Hf = 1.0; Rμ = 1.5; CAR = 1.0; Rpo = 1.5.
  • Fp = 0.4 · 1.5 · 1.5 · 4,000 · (1.0/1.5) · (1.0/1.5) = 1,600 lb.
  • Fp,min = 0.3 · 1.5 · 1.5 · 4,000 = 2,700 lb.
  • Fp,max = 1.6 · 1.5 · 1.5 · 4,000 = 14,400 lb.
  • Design Fp = 2,700 lb (minimum bound governs — equation result is rejected).

Worked example — flexible cooling tower, roof, retail

  • SDS = 1.0 g, Ip = 1.0, Wp = 6,000 lb.
  • z/h = 1.0 → Hf = 2.5; Rμ = 1.0; CAR = 2.5; Rpo = 1.5.
  • Fp = 0.4 · 1.0 · 1.0 · 6,000 · (2.5/1.0) · (2.5/1.5) = 10,000 lb.
  • Fp,max = 1.6 · 1.0 · 1.0 · 6,000 = 9,600 lb.
  • Design Fp = 9,600 lb (upper bound governs — equation result is capped).

Why the bounds exist (the engineering rationale)

  • Fp,min ensures that even the most ductile/forgiving combination still produces a meaningful seismic demand — a hospital cannot fall back on "the equation says 200 lb" for a 4,000 lb chiller.
  • Fp,max caps the response at the building's actual delivery capacity. The equation is a simplified envelope; the cap prevents over-amplification beyond what physics allows.

How to present the bounds on the calc cover sheet

  1. State SDS, Ip, Wp, Hf, Rμ, CAR, Rpo.
  2. Compute Fp, Fp,min, Fp,max all three.
  3. State which one governs.
  4. Use the governing value in the anchor design — and note it on the cover sheet.

Common mistakes

  • Computing only Fp and skipping the bounds entirely.
  • Computing Fp,min and Fp,max with the wrong Ip (e.g., using 1.0 in the minimum bound when the component is Ip = 1.5).
  • Reporting Fp from the equation as the design value when Fp,min governs — most common error on small floor-mounted equipment.
  • Reporting an unphysically large Fp when Fp,max would have capped it.

How PANACHE ENGINEERING handles this

Our calculator computes all three values and shows which governs. See our ASCE 7-22 Chapter 13 guide for the full force derivation, or request a stamped calculation.