Seismic Load Full Calculation Example — ASCE 7-22 ELF for 5-Story Steel Building

Step-by-step equivalent lateral force (ELF) procedure per ASCE 7-22 Section 12.8 for a 5-story steel special moment frame (SMF) office building in Los Angeles, California. This example covers the complete seismic load path from mapped spectral accelerations through base shear, vertical force distribution, story shears, overturning moments, and horizontal diaphragm design forces.

Related pages: Seismic Design Basics | Seismic Design Categories | Seismic Drift Guide | ASCE 7-22 Seismic Load Worked Example | Seismic Base Isolation Guide

Problem Statement

A 5-story steel office building in downtown Los Angeles, California:

Step 1 — Mapped Spectral Accelerations Ss and S1 (§22)

From ASCE 7-22 Figure 22-2 (Ss map) and Figure 22-3 (S1 map) for Los Angeles:

[ S_S = 2.40g \quad \text{(0.2 sec spectral response, Site Class B)} ] [ S_1 = 0.85g \quad \text{(1.0 sec spectral response, Site Class B)} ]

These are mapped Risk-Targeted Maximum Considered Earthquake (MCER) spectral accelerations for Site Class B (rock).

Step 2 — Site Coefficients Fa and Fv (§11.4.3)

From ASCE 7-22 Table 11.4-1 (Fa) and Table 11.4-2 (Fv) for Site Class D:

For Ss = 2.40g and Site Class D: Fa = 1.0 For S1 = 0.85g and Site Class D: Fv = 1.5

Note: ASCE 7-22 updated the site coefficients relative to ASCE 7-16. For sites with Ss ≥ 2.0, the maximum considered earthquake spectral response accelerations may be governed by deterministic limits in regions near active faults.

Step 3 — Design Spectral Accelerations SDS and SD1 (§11.4.4)

[ S_{DS} = \frac{2}{3} \times F_a \times S_S = \frac{2}{3} \times 1.0 \times 2.40 = 1.60g ]

[ S_{D1} = \frac{2}{3} \times F_v \times S_1 = \frac{2}{3} \times 1.5 \times 0.85 = 0.85g ]

Step 4 — Seismic Design Category (§11.6)

SDS = 1.60g ≥ 0.50g → SDC D for short-period category. SD1 = 0.85g ≥ 0.20g → SDC D for long-period category.

Seismic Design Category = D. This governs detailing requirements and height limits.

Step 5 — Fundamental Period T (§12.8.2)

From ASCE 7-22 Eq. 12.8-7 for steel moment frames:

[ T_a = C_t \times h_n^x = 0.028 \times (67)^{0.8} = 0.028 \times 26.4 = 0.74 \text{ sec} ]

Upper limit: Cu = 1.4 per Table 12.8-1 for SD1 ≥ 0.4. T_max = 1.4 × 0.74 = 1.04 sec. Use T = Ta = 0.74 sec (conservative lower bound).

Step 6 — Seismic Response Coefficient Cs (§12.8.1.1)

Limit 1 (short-period plateau):

[ Cs = \frac{S{DS}}{R/I_e} = \frac{1.60}{8/1.0} = 0.200 ]

Limit 2 (long-period descending branch):

[ Cs = \frac{S{D1}}{T \times (R/I_e)} = \frac{0.85}{0.74 \times 8} = \frac{0.85}{5.92} = 0.144 ]

Minimum Cs:

Cs,min = MAX[0.044 × 1.60 × 1.0, 0.01] = MAX[0.070, 0.01] = 0.070

Additionally, for S1 ≥ 0.6g: Cs,min ≥ (0.5 × 0.85)/(8/1.0) = 0.053.

Governing: Cs = MIN(0.200, 0.144) = 0.144. Exceeds the minimum of 0.070.

Step 7 — Seismic Base Shear V (§12.8.1)

Seismic weight by floor (dead load only per §12.7.2 — office live load ≤ 100 psf exempt from 20% inclusion):

Total effective seismic weight W = 5,000 kips.

[ V = 0.144 \times 5,000 = 720 \text{ kips} ]

Two perimeter SMF frames: each resists 360 kips.

Step 8 — Vertical Distribution of Seismic Forces (§12.8.3)

The exponent k accounts for the period-dependent mode shape:

• T ≤ 0.5 sec: k = 1 (linear)
• T ≥ 2.5 sec: k = 2 (parabolic)
• For T = 0.74 sec: k = 1.12 (interpolated)

The slight discrepancy between total Vx at floor 2 (722.1 kips) and base V (720.0 kips) is rounding.

Step 9 — Overturning Moment (§12.8.5)

Overturning moment at the base = sum of floor forces multiplied by their height above base:

M_base = 237.6 × 52 + 223.9 × 39 + 159.8 × 26 + 100.8 × 13 = 35,196 kip-ft

For a building width of 100 ft, the axial force couple in perimeter columns is ±352 kips — well within W14 column capacity.

Step 10 — Story Drift Check (§12.8.6)

Design story drift delta_x = Cd × delta_xe / Ie. For SMF: Cd = 5.5, Ie = 1.0.

Estimating elastic drift for a typical SMF:

At the 2nd floor, the design drift exceeds the limit by 2.2%. Increase column sizes or accept the slight exceedance with engineering judgment.

Step 11 — Horizontal Diaphragm Design Forces (§12.10.1.1)

Diaphragm forces with 0.32 wpx minimum (0.4wpx maximum):

Roof diaphragm: ELF-derived Fp5 = 237.6 kips = 0.297 wpx. Since 0.297 < 0.32 → Fp5 = 0.32 × 800 = 256 kips.

Floor diaphragms (Levels 2–4): ELF-derived values range 0.22–0.30 wpx, all below 0.32 minimum → Fp = 320 kips each.

The 320 kip minimum diaphragm force represents a uniform load of 320 kips / 100 ft = 3.2 klf across the diaphragm depth — easily accommodated by 3.5 in LW concrete on 3 in composite deck.

Step 12 — Accidental Torsion (§12.8.4.2)

Accidental torsion moment Mta = Fx × e_acc where e_acc = 0.05 × building dimension:

These torsion moments add approximately 2–5% force to the critical frame, within Omega0 overstrength capacity.

Key Takeaways

1. Cs controls design economy: At Cs = 0.144 (14.4% of building weight), the SMF system incurs moderate seismic demand. A braced frame (R = 6) would increase Cs to 0.267 — nearly doubling steel tonnage.

2. Diaphragm minimum forces often govern: The 0.32 wpx minimum increased diaphragm forces by 24–44% above ELF-derived values.

3. Drift check drives member sizes: Story drift at floor 2 marginally exceeds the 0.020hsx limit, meaning drift (not strength) controls frame design in SDC D.

4. Site class matters: Changing from Site Class D to E would drop Cs from 0.144 to about 0.128 — an 11% reduction while requiring additional geotechnical investigation.

Frequently Asked Questions

When should I use the ELF procedure vs. modal response spectrum analysis?

ELF is permitted for RC I/II buildings up to 160 ft with no plan or vertical irregularities per §12.6 and Table 12.6-1. Modal analysis is required for taller buildings, buildings with Type 1 horizontal irregularities, or when base isolation is used. For our 67 ft regular SMF building, ELF is both permitted and efficient.

How do I handle the P-Delta effect in ELF analysis?

Evaluate the stability coefficient theta = Px × delta × Ie / (Vx × hsx × Cd) per §12.8.7. If theta < 0.10, neglect P-Delta. If 0.10 ≤ theta ≤ theta_max, amplify forces by 1/(1−theta). If theta > theta_max, the structure is unstable. For SMF with Cd = 5.5, theta_max = 0.5/(1.0 × 5.5) = 0.091 — meaning drift near the 0.020hsx limit already pushes theta near the stability boundary.

What is the difference between the ELF base shear and the modal analysis base shear?

Per §12.9.1.4, modal base shear Vt must be ≥ 85% of ELF base shear V for regular buildings (100% for irregular). If Vt < 0.85V, scale all modal forces by 0.85V/Vt. This prevents unconservative results from modal truncation errors.

How does ASCE 7-22 differ from ASCE 7-16 for ELF?

ASCE 7-22 introduced multi-period response spectra replacing the two-point SDS/SD1 spectrum for certain structures. The design acceleration parameters now use T0 and TS transitions. For typical buildings under 100 ft, differences are minimal (< 3%). For tall buildings near source faults, the multi-period spectrum may produce notably different forces.