Natural Period of a 3-Story Steel Frame — Modal Analysis Worked Example

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Oct 21, 2026

A small three-story steel building analyzed for its natural periods, compared with the ASCE 7 approximate period, then changed in two ways: rotating some columns, and adding braces. The modal results explain why the first mode runs the "wrong" way, and how far an unbraced moment frame sits from the code's approximate period.

The building

Framing	2 × 2 bays at 25 ft, 3 stories at 12 ft (h_n = 36 ft), steel moment frames in both directions, fixed bases, rigid floor diaphragms
Sections	Columns: I-shape d = 12, b_f = 12, t_f = 0.6, t_w = 0.4 in (I_z = 510 in&sup4;, I_y = 173 in&sup4;) · Beams: d = 18, b_f = 7.5, t_f = 0.6, t_w = 0.36 in
Mass	Member self-weight plus 1.0 kip/ft dead load on every beam: total 963 kip

Model "3D building frame" with its defaults (2 × 2 bays, 3 stories), then open the Modes tab.

Open 3D frame analysis →

Deformed shape of a 3-story 2 by 2 bay steel frame under wind in the X direction — The model under service wind in +X (10 kip per floor), deformed shape scaled.

Step 1 · Modal results

Six modes, periods and mass participation

Mode	Period (s)	Mass X	Mass Z	Shape
1	1.165	0%	88.6%	1st translation, Z
2	0.896	0%	0%	1st torsion
3	0.836	85.9%	0%	1st translation, X
4	0.389	0%	9.4%	2nd translation, Z
5	0.285	0%	0%	2nd torsion
6	0.259	11.2%	0%	2nd translation, X

Two modes per direction reach 98% of the mass in Z (88.6 + 9.4) and 97.1% in X, past the 90% that ASCE 7 §12.9.1.1 requires for a modal response spectrum analysis.

Step 2 · Why Z is softer

Every column has its weak axis facing Z

Local y of a vertical member points along global +X, so the strong axis (I_z = 510 in&sup4;) resists X sway
and the weak axis (I_y = 173 in&sup4;) resists Z sway: about one third of the stiffness from every column.

Turning the three columns on the middle frame line by β = 90° balances the building:

Variant	T, Z mode (s)	T, X mode (s)	T, torsion (s)
All columns strong axis in X	1.165	0.836	0.896
Middle-line columns rotated 90°	0.974	0.912	0.838
X-braced end bays, both directions	0.199	0.196	0.143

Step 3 · Compare with the ASCE 7 approximate period

T_a = C_t h_n^x (ASCE 7-22 Eq. 12.8-8, Table 12.8-2)

Steel moment frame: C_t = 0.028, x = 0.8 → T_a = 0.028 × 36^0.8 = 0.492 s; C_uT_a = 1.4 × 0.492 = 0.689 s (S_D1 ≥ 0.4)
Braced frame ("all other"): C_t = 0.02, x = 0.75 → T_a = 0.02 × 36^0.75 = 0.294 s; C_uT_a = 0.412 s

The analytical periods of the moment frame (0.84–1.17 s) are well above C_uT_a. For the equivalent lateral force base shear, the period used may not exceed C_uT_a = 0.689 s (§12.8.2). The code doesn't trust a bare-frame model to be that flexible, because nonstructural walls, stairs and slab action stiffen real buildings. Drift, on the other hand, is computed with the analytical model (§12.8.6.2).

Mass is half the period. T grows with √(M/K). This model's mass is only the beam line loads and the self-weight. A real floor adds the slab, superimposed dead load, partitions and the cladding, often 70–100 psf over the whole floor area. That is roughly double the mass here and about 40% longer periods. Put the real seismic weight into the dead load case before you trust T.

Step 4 · Why the diaphragm matters

Same braced building, diaphragms off

With the floor diaphragms turned off, the braced building's first period jumps from 0.199 s to 1.150 s, with only 37% mass participation. Braces exist only on the outer frame lines, so without a slab tying the floor together, the unbraced middle frame line sways on its own. A rigid-diaphragm constraint makes each floor move as one body in plan, which is how a concrete-on-deck floor behaves. If your floor isn't rigid (plywood, metal deck without fill), the truth lies in between.

Tools used in this example

3D frame analysis (3D building generator, rigid diaphragms, modal), and the frame modeling guide for local axes, β rotations and mass.

Frequently asked questions

What is the approximate fundamental period of a steel moment frame?

ASCE 7-22 Eq. 12.8-8 gives Ta = Ct·hn^x with Ct = 0.028 and x = 0.8 for steel moment-resisting frames (hn in feet). For a 36 ft building that is 0.49 s. The period used to compute the base shear may not exceed Cu·Ta, where Cu = 1.4 when SD1 ≥ 0.4.

Why does a frame model give a longer period than ASCE 7?

A bare-frame model ignores the stiffness of walls, stairs, slabs and connections that real buildings have, and the approximate formula is calibrated to measured buildings. That is why ASCE 7 caps the period used for base shear at Cu·Ta, while drift is computed with the model.

What does a rigid diaphragm do in a 3D frame model?

It ties all the joints of a floor together so the floor translates and rotates in plan as a rigid body, while each joint keeps its own vertical displacement. Lateral load then goes to the frames in proportion to their stiffness, and torsion is resisted by the frames' lever arms about the center of rigidity.