Statics and Strength of Materials for Architecture and

Statics and Strength of Materials
Onouye
Kane
Fourth Edition
ISBN 978-1-29202-707-4
9 781292 027074
Statics and Strength of Materials for
Architecture and Building Construction
Barry S. Onouye Kevin Kane
Fourth Edition
Pearson Education Limited
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Essex CM20 2JE
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ISBN 10: 1-292-02707-X
ISBN 10: 1-269-37450-8
ISBN 13: 978-1-292-02707-4
ISBN 13: 978-1-269-37450-7
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Printed in the United States of America
Analysis of Selected Deter minate Str uctur al Systems
FBD (c)
FBD (d)
Using FBD (c),
C g MC = 0 D - 2.4 k112¿ 2 - 0.8 k14¿ 2 + 5 k118¿2
+3.44 k124¿ 2 - Ax 128¿ 2 = 0
28Ax = 140.56
‹ Ax = 5.02 k1;2
For the internal pin forces at C,
C g Fx = 0 D + 5 k - 5.02 k - 0.8 k + Cx = 0
‹ Cx = + 0.82 k
C g Fy = 0 D - 3.44 k + 2.4 k + Cy = 0
‹ Cy = + 1.04 k
Going back to FBD (b) of the entire frame, solve for Bx:
C g Fx = 0 D + 5 k - 5.02 k - 0.8 k + 0.8 k - Bx = 0
‹ Bx = - 0.02 k
The negative sign in the r esult for Bx indicates that the
original direction assumed in the FBD was incorrect.
‹ Bx = 0.02 k1:2
174
Analysis of Selected Deter minate Str uctur al Systems
Problems
Determine all support and pin for ces for the multifor ce
member diagrams listed below.
28
29
30
175
Analysis of Selected Deter minate Str uctur al Systems
31
32
33
176
Analysis of Selected Deter minate Str uctur al Systems
6 RETAINING WALLS
As the name implies, retaining walls are used to hold back
(retain) solid or other granular material to maintain a difference in ground elevation. A dam is a retaining wall used
to resist the lateral pressure of water or other fluids.
There are three general types of r etaining walls: (a) the
gravity wall (Figur e 76), (b) the r einforced concrete
cantilever retaining wall (Figure 77), and (c) the reinforced
concrete cantilever r etaining wall with counterforts
(Figure 78).
Gravity retaining walls are generally built of plain concrete or masonry. Height h is generally less than four feet
(1.3 m). A gravity wall depends on its mass to give it stability against the horizontal for ces from the soil. Sliding
resistance (friction) is developed between the concrete and
soil at the base. Some major dams are constructed as gravity wall systems, but understandably, the base dimensions
are immense.
Figure 76
Gravity retaining wall.
Figure 77
wall.
Reinforced cantilever retaining
Figure 78
Counterfort wall.
Reinforced concrete cantilever retaining walls are the most
frequently used type of retaining wall, with an effectiveness up to a height (h) of about 20 to 25 feet (6 to 7.6 m).
Stability of this wall type is achieved by the weight of the
structure and the weight of the soil on the heel of the slab
base. Sometimes a shear key is included at the bottom of
the slab base to incr ease the wall’s r esistance to sliding.
Retaining walls should have their foundations well below
the frost line, and adequate drainage (weep) holes near the
bottom of the wall should be provided to permit the water
accumulation behind the wall to escape.
As the height of a r etaining wall increases, the bending
moment in the cantilever wall incr eases, requiring more
thickness. The addition of counterforts (vertical triangular-shaped cross-walls) provides the additional depth at
the base to absorb the large bending stresses. Counterfort
walls behave like one-way slabs that span horizontally
between the counterforts. Counterforts are called buttresses
when this same configuration is used for the r etained
earth that is on the flat side of the wall.
Saturated loose sand or gravel, granular soil, or mud
cause pressures against retaining walls in a manner similar to true fluids (liquids) by exerting a horizontal pr essure. In true liquids, like water , the horizontal pr essure
and the vertical pr essure are the same at a given depth.
However, in soil, the horizontal pressure is less than the
vertical pressure, with the ratio dependent on the physical
177
Analysis of Selected Deter minate Str uctur al Systems
properties of the soil. Soil pr essure, as with liquids,
increases proportionately with its depth below grade
(Figure 79).
Lateral pressure increases linearly from zero at the top to
a maximum at the bottom of the footing.
p = ω¿ * h
where
p = the magnitude of the earth pressure in psf or 1kN>m2 2
Figure 79
FBD of a gravity retaining wall.
ω¿ = the “equivalent” fluid weight (density) of the
soil in pounds per cubic feet. V alues range from a
minimum of 30 pcf (for well-graded, clean gravelsand mixes) to 60 pcf (for clayey sands). SI values are
4.7 to 9.4 kN>m3.
h = soil depth in feet 1m2.
Therefore,
P =
1
* 1pmax * h2 * 1 ft. or 1 m
2
where P = the lateral force (pounds, kips, N, or kN) based
on the area of the pressure distribution acting on a 1-footwide (1-m-wide) strip of wall.
pmax = the maximum pressure at depth h (psf or kN/m2)
Equivalent fluid pressure against a retaining wall may create conditions of instability. Retaining walls are susceptible
to three failure modes: (a) sliding—when the friction at the
footing base is insufficient to resist sliding; (b) overturning
about the toe—when the lateral for ce produces an overturning moment greater than the stabilizing moment from
the wall’s weight, slab base weight, and the soil mass
above the heel; and (c) excessive bearing pr essure at the
toe—when the combination of the vertical downward force
and the compression at the toe caused by the horizontal
force exceeds the allowable bearing pressure of the soil.
The pressure distribution under the base (Figur e 80) depends upon the location and magnitude of the r esultant
(vertical and horizontal forces) force as it passes through
the footing base.
Figure 80
footing.
178
Bearing pressure under the wall
Analysis of a cantilever r etaining wall requires that the
equilibrium summation of moments about the toe is stable; that is, the weight of the wall plus the backfill on the
heel exceeds the overturning moment of the active soil
pressure by a factor of at least 1.5 (a safety factor imposed
by building codes). Once a stable configuration is achieved,
the soil pressure distribution on the footing must be calculated to ensur e that the bearing pr essures are within
allowable limits for the soil on site.