Lateral loads: wind and earthquakes

The fourth class meets on the roof of the school on a gusty February morning. Szymańska hands out a small anemometer and asks Ingrid to read the wind speed at the parapet, then at the middle of the roof, then crouched behind the elevator penthouse. The readings jump from strong to gentle to almost nothing within a few meters. 'Gravity is polite,' Szymańska says. 'It always pulls the same way and with the same force. Wind is rude. It comes from every direction, it gusts, it sucks as well as pushes, and it changes with height and with whatever is upwind. Earthquakes are worse: they do not push on the building at all. They shake the ground and let the building's own weight do the damage.'

How wind loads a building

Moving air has kinetic energy, and when a building stops it, that energy becomes pressure. The basic quantity is the velocity pressure, which is proportional to the density of air and the square of the wind speed. Doubling the wind speed quadruples the pressure, which is why small differences in design wind speed matter. Codes give a reference velocity pressure or a basic wind speed for each location, based on the strongest winds expected over a long return period, and the designer adjusts it for height, terrain, gusts and the shape of the building.

Wind speed increases with height above the ground, because friction slows the air near the surface. Over open country the increase is rapid near the ground; in a city center, the buildings slow the lower air further and the increase is spread over a greater height. Codes describe this with an exposure or height factor that grows with height and depends on the roughness of the terrain upwind. Wind also gusts, and the peaks of short gusts are what a building feels; a gust factor raises the mean pressure to account for them.

The pressure on each face of a building depends on its shape. On the windward face the air is stopped and pushes inward. On the leeward face, and on the side walls and most of the roof, the air flowing around the building creates suction, pulling outward. Pressure coefficients, given by codes from wind tunnel tests, describe these effects: a typical rectangular building has a windward coefficient of about +0.8 and a leeward coefficient of about −0.5. For the structure as a whole, the windward push and leeward pull add together. Corners and roof edges, where the flow separates, see much higher local suctions, which govern the design of cladding, roofing and their fastenings, though not of the main structure.

ASCE 7-22 calculates the velocity pressure as qz = 0.00256 Kz Kzt Ke V², in pounds per square foot, where V is the basic wind speed in miles per hour from the code's maps, and Kz, Kzt and Ke account for height and exposure, topography and elevation. The design pressure on the main structure is then p = qz Kd G Cp, with the directionality factor Kd and the gust effect factor G. For Minneapolis this example takes a basic wind speed of 107 mph for an ordinary building.

How earthquakes load a building

An earthquake moves the ground under a building back and forth, and the building's inertia resists the motion. The forces that result are proportional to the building's mass and to the accelerations it experiences, so a heavy building attracts larger earthquake forces than a light one. How strongly a building responds depends on its natural period, the time it takes to sway back and forth once. Stiff, low buildings have short periods, a tenth to a half of a second; tall, flexible buildings have periods of several seconds. Ground shaking is strongest in a certain range of periods, so buildings in that range respond most strongly.

Seismic hazard varies enormously across North America. The West Coast, Alaska, the Intermountain West, the New Madrid region around Memphis and parts of the Southeast have significant hazard; the upper Midwest, including Minneapolis, has very low hazard. Codes give a design spectrum for each location: the spectral acceleration, the peak acceleration a simple structure would feel, for a range of periods. The local soil also matters: soft soils amplify shaking, so the spectrum is adjusted for the site class.

Most buildings are designed by an equivalent static method: the code's base shear, a horizontal force at the foundation, is calculated from the spectral acceleration at the building's period, the building's weight and factors for its importance and its structural system. That force is then distributed up the height of the building in proportion to the weight and height of each floor, so the upper floors receive the largest forces, as they would in the building's first mode of swaying. In ASCE 7, the base shear is V = Cs W, where the seismic response coefficient Cs is the design spectral acceleration SDS divided by R/Ie, with R the response modification factor of the structural system and Ie the importance factor, subject to minimum values.

The division by a ductility factor is the most surprising idea in seismic design. Codes do not expect buildings to stay elastic in a rare, large earthquake; that would make them impossibly strong and expensive. Instead, they let ductile structures yield and dissipate energy in controlled places, like a paper clip bent back and forth, and reduce the design forces accordingly. A very ductile system, such as a carefully detailed steel moment frame or wood shear walls with many nails, gets a large reduction; a brittle system, such as unreinforced masonry, gets almost none. The price is detailing: the ductile places must be designed to bend without breaking, and everything else must be strong enough to let them.

Greenway Homes: wind or earthquake?

Szymańska asks the students to calculate both lateral loads for the east bar of Greenway Homes, the four-floor wood-frame building that Wren is developing in her housing studio. The bar is 92 ft long, 56 ft deep and 40 ft tall. The script below finds the wind pressures and forces at each floor for wind on the long face, then the seismic base shear for two sites: the actual site, and a hypothetical copy of the building in a high-hazard city.

# Wind and earthquake forces on the east bar of Greenway Homes (ASCE 7-22 directional procedure, simplified)
floors, fh = 4, 10                   # number of floors and floor-to-floor height (ft)
Lw, Dw = 92, 56                    # face exposed to wind and depth of the building (ft)
V_mph, Kd, G = 107, 0.85, 0.85             # basic wind speed, directionality and gust factors
Kz = lambda z: 2.01 * (max(z, 30) / 1200) ** (2 / 7.0)   # exposure B
q = lambda z: 0.00256 * Kz(z) * V_mph ** 2
Cp_w, Cp_l = 0.8, -0.5
p_wind = lambda z: q(z) * Kd * G * Cp_w
p_lee = q(floors * fh) * Kd * G * -Cp_l
print("Level  height   windward  leeward   force at level")
total_w = 0.0
for i in range(1, floors + 1):
    z = i * fh
    trib = fh if i < floors else fh / 2
    pw, pl = p_wind(z), p_lee
    F = (pw + pl) * trib * Lw / 1000
    total_w += F
    print(f"  {i}    {z:5.1f}   {pw:7.1f}   {pl:7.1f}   {F:7.1f} kips")
print(f"Wind base shear {total_w:.0f} kips (specified), {1.0 * total_w:.0f} kips factored")
W = [200] * (floors - 1) + [160]         # seismic weight of each level, top level is the roof (kips)
hs = [fh * i for i in range(1, floors + 1)]
for city, Sa in (("Minneapolis", 0.07), ("Seattle", 1.0)):
    Cs = max(Sa / 6.5, 0.044 * Sa, 0.01)          # SDS / (R / Ie), wood structural panel shear walls
    V = Cs * sum(W)
    sw = sum(w * hh for w, hh in zip(W, hs))
    Fx = [V * w * hh / sw for w, hh in zip(W, hs)]
    print(f"{city}: seismic base shear {V:.0f} kips = {100 * V / sum(W):.1f} % of the weight; level forces "
          + ", ".join(f"{f:.0f}" for f in Fx))

Level  height   windward  leeward   force at level
  1     10.0      11.9       8.1      18.3 kips
  2     20.0      11.9       8.1      18.3 kips
  3     30.0      11.9       8.1      18.3 kips
  4     40.0      12.9       8.1       9.6 kips
Wind base shear 65 kips (specified), 65 kips factored
Minneapolis: seismic base shear 8 kips = 1.1 % of the weight; level forces 1, 2, 3, 3
Seattle: seismic base shear 117 kips = 15.4 % of the weight; level forces 13, 25, 38, 41

The results are striking. For the real site, the factored wind base shear on the long face is about 65 kips, while the seismic base shear is only about 8 kips, about a sixth of the wind load or less. In Minneapolis, wind governs the lateral design of almost every building. Moved to Seattle, the same building would attract a seismic base shear of about 117 kips, more than the wind load and about a sixth of its weight. Callum asks why the light wood building attracts so much: the answer is that the spectral acceleration at its short period is high on the coast, and even a light building is heavy, with floors, walls, finishes and roofing adding up to several hundred kips.

The comparison also shows why seismic design is about weight. A concrete version of the same building, about four times heavier as Lesson 2 estimated, would attract about four times the seismic force, while the wind force, which depends only on the building's shape and size, would be the same. In seismic regions, light structures have a real advantage, and heavy features such as rooftop gardens, thick masonry cladding and concrete toppings carry a seismic cost.

Configuration

How a building is shaped and how its lateral system is arranged matter as much as the size of the forces. Codes penalize irregularities, because they concentrate damage. A soft story, a floor much less stiff than the floors above, often a ground floor of open shops or parking under apartments, concentrates all the drift in one level; many buildings have collapsed this way, including apartments over parking in the 1994 Northridge earthquake in California. A torsional irregularity, where the stiffness is far from the center of mass, makes the building twist. Re-entrant corners in L-shaped, T-shaped or U-shaped plans concentrate stress where the wings meet. Discontinuous walls that stop above a large opening, and sudden changes of mass, such as a heavy pool on an upper floor, are also irregularities.

Greenway Homes is U-shaped, with its north bar, east bar and townhouses meeting at corners. Szymańska suggests that in a seismic region the engineer would probably separate the wings with seismic joints, gaps wide enough that the wings can sway independently without hammering each other, so that each wing is a simple rectangle. In Minneapolis, where wind governs and seismic forces are small, the wings can be tied together, but the corners must still be detailed to transfer forces between the wings. The architect's part is to make the joints and the lateral system part of the plan from the start: stair and elevator cores placed near the ends and middle of each wing, walls that line up from floor to floor, and no soft story under the homes.

Drift, damping and comfort

Both wind and earthquakes make buildings sway. For earthquakes, codes limit interstory drift, typically to about 1/40 to 1/50 of the story height for ordinary buildings, to limit damage and prevent instability. For wind, designers limit drift to protect cladding and partitions, and for tall buildings they also limit accelerations, because people feel motion long before it is dangerous. Tall, slender towers are tested in wind tunnels, and some carry tuned mass dampers near the top, heavy masses on springs or pendulums that move against the building's sway. These matter mostly for towers of thirty floors or more; for a four-floor building like Greenway, strength and simple drift checks are enough.

Exercise 4: lateral loads

• Find the reference wind pressure or basic wind speed for your hometown in the code and compare it with Minneapolis.
• Estimate the wind base shear on the long face of your studio project with a net pressure of 20 psf.
• Estimate the weight of your building per floor and the seismic base shear for a low-hazard and a high-hazard city.
• Mark any irregularities in your plan and section: soft floors, re-entrant corners, walls that stop above openings.

Self-check

• Can I explain why wind pressure grows with the square of the wind speed?
• Do I know why heavier buildings attract larger earthquake forces?
• Can I identify a soft story and explain why it is dangerous?