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5.
SECTION FIVE
STRUCTURAL THEORY
Akbar Tamboli, Michael Xing, Mohsin Ahmed
Thornton-Tomasetti Engineers, Newark, New Jersey
STRUCTURAL THEORY CREATES IDEALIZATION
OF STRUCTURE FOR PURPOSES OF ANALYSIS
Structural modeling is an essential and important tool in structural engineering. Over the past 200 years, many of the most significant contributions to the under- standing of the structures have been made by Scientist Engineers while working on mathematical models, which were used for real structures. Application of mathematical model of any sort to any real structural system must be idealized in some fashion; that is, an analytical model must be developed. There has never been an analytical model, which is a precise representation of the physical system. While the performance of the structure is the result of natural effects, the development and thus the performance of the model is entirely under the control of the analyst. The validity of the results obtained from applying math- ematical theory to the study of the model therefore rests on the accuracy of the model. While this is true, it does not mean that all analytical models must be elaborate, conceptually sophisticated devices. In some cases very simple models give surprisingly accurate results. While in some other cases they may yield an- swers, which deviate markedly from the true physical behavior of the model, yet be completely satisfactory for the problem at hand. Structure design is the application of structural theory to ensure that buildings and other structures are built to support all loads and resist all constraining forces that may be reasonably expected to be imposed on them during their expected service life, without hazard to occupants or users and preferably without dangerous deformations, excessive sideways (drift), or annoying vibrations. In addition, good design requires that this objective be achieved economically. Provision should be made in application of structural theory to design for ab- normal as well as normal service conditions. Abnormal conditions may arise as a result of accidents, fire, explosions, tornadoes, severer-than-anticipated earthquakes, floods, and inadvertent or even deliberate overloading of building components. Un- der such conditions, parts of a building may be damaged. The structural system, however, should be so designed that the damage will be limited in extent and undamaged portions of the building will remain stable. For the purpose, structural elements should be proportioned and arranged to form a stable system under normal
5.2 SECTION FIVE
service conditions. In addition, the system should have sufficient continuity and ductility, or energy-absorption capacity, so that if any small portion of it should sustain damage, other parts will transfer loads (at least until repairs can be made) to remaining structural components capable of transmitting the loads to the ground. (‘‘Steel Design Handbook, LRFD Method’’, Akbar R. Tamboli Ed., McGraw- Hill 1997. ‘‘Design Methods for Reducing the Risk of Progressive Collapse in Buildings’’. NBS Buildings Science Series 98, National Institute of Standards and Technology, 1997. ‘‘Handbook of Structural Steel Connection Design and Details’’, Akbar R. Tamboli Ed., McGraw-Hill 1999’’).
5.1 DESIGN LOADS
Loads are the external forces acting on a structure. Stresses are the internal forces that resist them. Depending on that manner in which the loads are applied, they tend to deform the structure and its components—tensile forces tend to stretch, compressive forces to squeeze together, torsional forces to twist, and shearing forces to slide parts of the structure past each other.
5.1.1 Types of Loads
External loads on a structure may be classified in several different ways. In one classification, they may be considered as static or dynamic. Static loads are forces that are applied slowly and then remain nearly constant. One example is the weight, or dead load, of a floor or roof system. Dynamic loads vary with time. They include repeated and impact loads. Repeated loads are forces that are applied a number of times, causing a variation in the magnitude, and sometimes also in the sense, of the internal forces. A good example is an off-balance motor. Impact loads are forces that require a structure or its components to absorb energy in a short interval of time. An example is the dropping of a heavy weight on a floor slab, or the shock wave from an explosion striking the walls and roof of a building. External forces may also be classified as distributed and concentrated. Uniformly distributed loads are forces that are, or for practical purposes may be considered, constant over a surface area of the supporting member. Dead weight of a rolled-steel I beam is a good example. Concentrated loads are forces that have such a small contact area as to be negligible compared with the entire surface area of the supporting member. A beam supported on a girder, for example, may be considered, for all practical purposes, a concentrated load on the girder. Another common classification for external forces labels them axial, eccentric, and torsional. An axial load is a force whose resultant passes through the centroid of a section under consideration and is perpendicular to the plane of the section. An eccentric load is a force perpendicular to the plane of the section under consideration but not passing through the centroid of the section, thus bending the supporting member (see Arts. 5.4.2, 5.5.17, and 5.5.19).
TABLE 5.1 5.
Minimum Design Dead Loads
Walls
Clay brick
High-absorption, per 4-in wytheMedium-absorption, per 4-in wytheLow-absorption, per 4-in wythe Sand-lime brick, per 4-in wytheConcrete brick
4-in, with heavy aggregate4-in, with light aggregate Concrete block, hollow
8-in, with heavy aggregate
lb / ft
2 34394638 4633 55
Floor Finishes
lb / ft
2
Asphalt block, 2-in
24
Cement, 1-in
12
Ceramic or quarry tile, 1-in
12
Hardwood flooring,
(^7) ⁄8-in
4
Plywood subflooring,
(^1) ⁄2-in
Resilient flooring, such as asphalt tile and linoleum
2
Slate, 1-in
15
Softwood subflooring, per in of thickness
3
Terrazzo, 1-in
13
Wood block, 3-in
4
8-in, with light aggregate12-in, with heavy aggregate12-in, with light aggregate Clay tile, loadbearing
4-in8-in12-in Clay tile, nonloadbearing
2-in4-in8-in Furring tile
358555 244258 111834
Wood joists, double wood floor, joist size
2
6 2
8 2
10 2
12 3
6 3
8 3
10 3
12 3
14
lb / ft
2
12-in spacing
(^66787891112)
16-in spacing
(^5667678910)
1 1 ⁄^2
-in 2-in Glass block, 4-inGypsum block, hollow
2-in4-in6-in
8 1018 9.512.518.
Concrete Slabs
lb / ft
2
Stone aggregate, reinforced, per in of thickness
Slag, reinforced, per in of thickness
Lightweight aggregate, reinforced, per in of thickness
6 to 10
TABLE 5.1 5.
Minimum Design Dead Loads (
Continued
)
Masonry
Cast-stone masonryConcrete, stone aggregate, reinforcedAshlar:
GraniteLimestone, crystallineLimestone, oo
¨litic
MarbleSandstone
Roof and Wall Coverings
Clay tile shinglesAsphalt shinglesComposition:
3-ply ready roofing4-ply felt and gravel5-ply felt and gravel Copper or tinCorrugated steelSheathing (gypsum),
(^1) ⁄^2 -in
Sheathing (wood), per in thicknessSlate,
(^1) ⁄4-in
Wood shingles Waterproofing
Five-ply membrane Ceilings
Plaster (on tile or concrete)Suspended metal lath and gypsum plasterSuspended metal lath and cement plasterSuspended steel channel supportsGypsumboard per
(^1) ⁄^4 -in thickness
lb / ft
3
144150 165165135173144 lb / ft
2
9 to 14
2 1 5.5 61 22 3 10 2 lb / ft
2 5 lb / ft
2 5 1015 2 1.
Floor Fill
lb / ft
2
Cinders, no cement, per in of thickness
5
Cinders, with cement, per in of thickness
9
Sand, per in of thickness
8
Partitions
lb / ft
2
Plaster on masonry
Gypsum, with sand, per in of thickness
Gypsum, with lightweight aggregate, per in
4
Cement, with sand, per in of thickness
10
Cement, with lightweight aggregate, per in
5
Plaster, 2-in solid
20
Metal studs
Plastered two sides
18
Gypsumboard each side
6
Wood studs, 2
4-in
Unplastered
3
Plastered one side
11
Plastered two sides
19
Gypsumboard each side
7
Glass
lb / ft
2
Single-strength
Double-strength
Plate,
(^1) ⁄^8 -in
Insulation
lb / ft
2
Cork, per in of thickness
Foamed glass, per in of thickness
Glass-fiber bats, per in of thickness
Polystyrene, per in of thickness
Urethane
Vermiculite, loose fill, per in of thickness
STRUCTURAL THEORY 5.
TABLE 5.2 Minimum Design Live Loads ( Continued )
b. Concentrated live loads d Location Load, lb
Elevator machine room grating (on 4-in^2 area) 300 Finish, light floor-plate construction (on 1-in^2 area) 200 Garages: Passenger cars: Manual parking (on 20-in^2 area) 2, Mechanical parking (no slab), per wheel 1, Trucks, buses (on 20-in^2 area), per wheel 16, Manufacturing Light 2, Heavy 3, Office floors (on area 2.5 ft square) 2, Scuttles, skylight ribs, and accessible ceilings (on area 2.5 ft square) 200 Sidewalks (on area 2.5 ft square) 8, Stair treads (on 4-in^2 area at center of tread) 300 Libraries (on area 2.5 ft square) 1, Hospitals (on area 2.5 ft square) 1, Schools (on area 2.5 ft square) 1, Stores (on area 2.5 ft square) 3,
d (^) Use instead of uniformly distributed live load, except for roof trusses, if concentrated loads produce greater stresses or deflections. Add impact factor for machinery and moving loads: 100% for elevators, 20% for light machines, 50% for reciprocating machines, 33% for floor or balcony hangers. For craneways, and a vertical force equal to 25% of maximum wheel load; a lateral force equal to 10% of the weight of trolley and lifted load, at the top of each rail; and a longitudinal force equal to 10% of maximum wheel loads, acting at top of rail.
Live Loads. These may be concentrated or distributed loads and should be con- sidered placed on the building to produce maximum effects on the structural mem- ber being designed. Minimum live loads to be used in building design are listed in Table 5.2. These include an allowance for impact, except as noted in the footnote of Table 5.2 b. Partitions generally are considered to be live loads, because they may be installed at any time, almost anywhere, to subdivide interior spaces, or may be shifted from original places to other places in the future. Consequently, unless a floor is designed for a large live load, for example, 80 lb / ft^2 , the weight of partitions should be added to other live loads, whether or not partitions are shown on the working drawings for building construction. Because of the low probability that a large floor area contributing load to a specific structural member will be completely loaded with maximum design live loads, building codes generally permit these loads to be reduced for certain types of occupancy. Usually, however, codes do not permit any reduction for places of public assembly, dwellings, garages for trucks and buses, or one-way slabs. For areas with a minimum required live load exceeding 100 lb / ft^2 and for passenger- car garages, live loads on columns supporting more than one floor may be decreased 20%. Except for the preceding cases, a reduced live load L , lb / ft^2 , may be computed from
5.8 SECTION FIVE
TABLE 5.2 Minimum Design Live Loads ( Continued )
c. Minimum design loads for materials
Material
Load, lb / ft^3 Material
Load, lb / ft^3
Aluminum, cast Bituminous products: Asphalt Petroleum, gasoline Pitch Tar Brass, cast Bronze, 8 to 14% tin Cement, portland, loose Cement, portland, set Cinders, dry, in bulk Coal, anthracite, piled Coal, bituminous or lignite, piled Coal, peat, dry, piled Charcoal Copper Earth (not submerged): Clay, dry Clay, damp Clay and gravel, dry Silt, moist, loose Silt, moist, packed Sand and gravel, dry, loose Sand and gravel, dry, packed Sand and gravel, wet Gold, solid
165
81 42 69 75 534 509 90 183 45 52 47 23 12 556
63 110 100 78 96 100 110 120 1205
Gravel, dry Gypspum, loose Ice Iron, cast Lead Lime, hydrated, loose Lime, hydrated, compacted Magnesium alloys Mortar, hardened; Cement Lime Riprap (not submerged): Limestone Sandstone Sand, clean and dry Sand, river, dry Silver Steel Stone, ashlar: Basalt, granite, gneiss Limestone, marble, quartz Sandstone Shale, slate Tin, cast Water, fresh Water, sea
104 70
450 710 32 45 112
130 110
83 90 90 106 656 490
165 160 140 155 459
64
L 0.25 L o (5.1)
AI
where Lo unreduced live load, lb / ft^2 (see Table 5.1 a ) AI ^ influence area, or floor area over which the influence surface for struc- tural effects is significantly different from zero area of four surrounding bays for an interior column, plus similar area from supported floors above, if any area of two adjoining bays for an interior girder or for an edge column, plus similar areas from supported floors above, if any area of one bay for an edge girder or for a corner column, plus similar areas from supported floors above, if any
The reduced live load L , however, should not be less than 0.5 L (^) o for members supporting one floor or 0.4 L (^) o for members supporting two or ore floors. Roofs used for promenades should be designed for a minimum life load of 60 lb / ft^2 , and those used for gardens or assembly, for 100 lb / ft 2. Ordinary roofs should be designed for a minimum live load L , lb / ft^2 , computed from
5.10 SECTION FIVE
Exposure A applies to centers of large cities, where for at least one-half mile upwind from the building the majority of structures are over 70 ft high and lower buildings extend at least one more mile upwind. Exposure B applies to wooded or suburban terrain or to urban areas with closely spaced buildings mostly less than 70 ft high, where such conditions prevail upwind for a distance from the building of at least 1500 ft or 10 times the building height. Exposure C exists for flat, open country or exposed terrain with obstructions less than 30 ft high. Exposure D applies to flat unobstructed areas exposed to wind blowing over a large expanse of water with a shoreline at a distance from the building or not more than 1500 ft or 10 times the building height. For design purposes also, the following formulas may be used to determine, for heights z (in feet) greater than 15 ft above ground, a pressure coefficient K for converting wind speeds to pressures. For Exposure A, for heights up to 1500 ft above ground level, 2 / 3 z
K 0.000517 (5.3)
For z less than 15 ft, K 0.00031. For Exposure B, for heights up to 1200 ft above ground level, 4 / 9 z
K 0.00133 (5.4)
For z less than 15 ft, K 0.00095. For Exposure C, for heights up to 900 ft above ground level, 2 / 7 z
K 0.00256 (5.5)
For z less than 15 ft, K 0.0020. For Exposure D, for heights up to 700 ft above ground level, 1 / 5 z
K 0.00357 (5.6)
For z less than 15 ft, K 0.0031. For ordinary buildings not subject to hurricanes, the velocity pressure qz , psf, at height z may be calculated from
q KV^2 (5.7) z
where V basic wind speed, mi / hr, but not less than 70 mi / hr. For important buildings, such as hospitals and communication buildings, for buildings sensitive to wind, such as slender skyscrapers, and for buildings present- ing a high degree of hazard to life and property, such as auditoriums, q (^) z computed from Eq. (5.7) should be increased 15%. To allow for hurricanes, qz should be increased 10% for ordinary buildings and 20% for important, wind-sensitive or high-risk buildings along coastlines. These increases may be assumed to reduce uniformly with distance from the shore to zero for ordinary buildings and 15% for the more important or sensitive buildings at points 100 mi inland.
STRUCTURAL THEORY 5.
Wind pressures on low buildings are different at a specific elevation from those on tall buildings. Hence, building codes may give different formulas for pressures for the two types of construction. In any case, however, design wind pressure should be a minimum of 10 psf.
Multistory Buildings. For design of the main wind-force resisting system of or- dinary, rectangular, multistory buildings, the design pressure at any height z , ft, above ground may be computed from
p (^) zw G Co pw q (^) z (5.8)
where p (^) zw design wind pressure, psf, on windward wall Go gust response factor Cpw external pressure coefficient q (^) z velocity pressure computed from Eq. (5.7) and modified for hurricanes and building importance, risks, and wind sensitivity
For windward walls, Cpw may be taken as 0.8. For side walls, Cpw may be assumed as 0.7 (suction). For roofs and leeward walls, the design pressure at elevation z is
p (^) zl ^ G C qo p h (5.9)
where p (^) zl design pressure, psf, on roof or leeward wall Cp external pressure coefficient for roof or leeward wall q (^) h velocity pressure at mean roof height h (see Fig. 3.1 d )
In these equations, the gust response factor may be taken approximately as
8.58 D G (^) o 0.65 (^) n 1 (5.10) ( h / 30)
where D 0.16 for Exposure A, 0.10 for Exposure B, 0.07 for Exposure C, and 0.05 for Exposure D n 1 ⁄ 3 for Exposure A, 2 ⁄ 9 for Exposure B, 1 ⁄ 7 for Exposure C, and 0.1 for Exposure D h mean roof height, ft
For leeward walls, subjected to suction, Cp depends on the ratio of the depth d to width b of the building and may be assumed as follows:
d / b 1 or less 2 4 or more
C (^) p 0.5 0.3 0.
The negative sign indicates suction. Table 5.3 lists values of Cp for pressures on roofs.
Flexible Buildings. These are structures with a fundamental natural frequency less than 1 Hz or with a ratio of height to least horizontal dimension (measured at mid-height for buildings with tapers or setbacks) exceeding 5. For such buildings, the main wind-force resisting system should be designed for a pressure on wind- ward walls at any height z , ft, above ground computed from
STRUCTURAL THEORY 5.
In ASCE-7-95 and 98, the basic wind speed changed from fast mile wind to 3- second gust wind speed in miles per hour. The wind speed values on the basic wind speed map has changed. This change should not have any big impact on the wind pressure. However, confusion is easily created because all the major building codes including the IBC 2000 are still using old basic wind speed map based on fast mile wind, and they repeatedly refer to ASCE-7 95 or 98. It is to be noted that the reference from the building codes to the ASCE-7 are either adoption of ASCE- 7 as an alternative approach or for certain factors that are not related to the basic wind pressure. In ASCE-7-95 and 98, new factors such as wind directionality factor, topo- graphic factor were introduced, and gust effect factors were updated for rigid struc- tures as well as for flexible / dynamically sensitive structures. The calculation be- came much more complicated than the approach in this book and the results should be more accurate. We suggest that for complicated structures it is necessary to use ASCE-7-98 method to check the results.
Snow, Ice, and Rain Loads. These, in effect, are nonuniformly distributed, ver- tical, live loads that are imposed by nature and hence are generally uncertain in magnitude and duration. They may occur alone or in combination. Design snow loads preferably should be determined for the site of the proposed building with the advice of meteorologists and application of extreme-value statistical analysis to rain and snow records for the locality. Rain loads depend on drainage and may become large enough to cause roof failure when drainage is blocked (see Art. 3.4.3). Ice loads are created when snow melts, then freezes, or when rain follows a snow storm and freezes. These loads should be considered in determining the design snow load. Snow loads may consist of pure snow or a mixture of snow, ice, and water. Design snow loads on roofs may be assumed to be proportional to the maximum ground snow load p (^) g , lb / ft^2 , measured in the vicinity of the building with a 50- year mean recurrence interval. Determination of the constant of proportionality should take into account:
1. Appropriate mean recurrence interval. 2. Roof exposure. Wind may blow snow off the roof or onto the roof from nearby higher roofs or create nonuniform distribution of snow. 3. Roof thermal conditions. Heat escaping through the roof melts the snow. If the water can drain off, the snow load decreases. Also, for sloped roofs, if they are warm, there is a tendency for snow to slide off. Insulated roofs, however, restrict heat loss from the interior and therefore are subjected to larger snow loads. 4. Type of occupancy and uses of building. More conservative loading should be used for public-assembly buildings, because of the risk of great loss of life and injury to occupants if overloads should cause the roof to collapse. 5. Roof slope. The steeper a roof, the greater is the likelihood of good drainage and that show will slide off.
In addition, roof design should take into account not only the design snow load uniformly distributed over the whole roof area but also possible unbalanced loading. Snow may be blown off part of the roof, and snow drifts may pile up over a portion of the roof.
5.14 SECTION FIVE
For flat roofs, in the absence of building-code requirements, the basic snow load when the ground snow load pg is 20 lb / ft^2 or less may be taken as
P (^) min p (^) g (5.15)
When pg is between 20 and 25 lb / ft^2 , the minimum allowable design load is p min 20 lb / ft^2 , and when p (^) g exceeds 25 lb / ft^2 , the basic snow load may be taken as
p (^) ƒ 0.8 p (^) g (5.16)
where p ƒ design snow load, lb / ft^2 , for a flat roof that may have unheated space underneath and that may be located where the wind cannot be relied on to blow snow off, because of nearby higher structures or trees p (^) g ground snow load, lb / ft^2
For roofs sheltered from the wind, increase p ƒ computed from Eq. (5.16) by 20%, and for windy sites, reduce p ƒ 10%. For a poorly insulated roof with heated space underneath, decrease p ƒ by 30%. Increase p ƒ 10% for large office buildings and public-assembly buildings, such as auditoriums, schools, factories. Increase p ƒ 20% for essential buildings, such as hospitals, communication buildings, police and fire stations, power plants, and for structures housing expensive objects or equipment. Decrease p. ƒ 20% for structures with low human occupancy, such as farm buildings. The ground snow load pg should be determined from an analysis of snow depths recorded at or near the site of the proposed building. For a rough estimate in the absence of building-code requirements, p (^) g may be taken as follows for the United States, except for mountainous regions:
0–5 lb / ft^2 —southern states from about latitude N32 southward 10–15 lb / ft^2 —Pacific coast between latitudes N32 and N40 and other states between latitudes N32 and N37 20–30 lb / ft^2 —Pacific coast from latitude N40 northward and other states between latitudes N37 and N40 40–50 lb / ft^2 —north Atlantic and central states between latitudes N40 and N43 60–80 lb / ft^2 —northern New England between latitudes N43 and N45 and cen- tral states from N43 northward 80–120 lb / ft^2 —Maine above latitude N45
For sloping roofs, the snow load depends on whether the roof will be warm or cold. In either case, the load may be assumed to be zero for roofs making an angle of 70 or more with the horizontal. Also, for any slope, the load need not be taken greater than p ƒ given by Eq. (5.16). For slopes , deg, between 0 and 70, the snow load, lb / ft^2 , acting vertically on the projection of the roof on a horizontal plane, may be computed for warm roofs from
70
p s p ƒ p ƒ (5.17)
and for cold roofs from
70
p s p ƒ p ƒ (5.18)
Hip and gable roofs should be designed for the condition of the whole roof
5.16 SECTION FIVE
Seismic Loads. These are the result of horizontal and vertical movements imposed on a building by earth vibrations during an earthquake. Changing accelerations of the building mass during the temblor create changing inertial forces. These are assumed in building design to act as seismic loads at the various floor and roof levels in proportion to the portion of the building mass at those levels. Because analysis of building response to such dynamic loading generally is very complex, building codes permit, for design of ordinary buildings, substitution of equivalent static loading for the dynamic loading (see Art. 5.18.6). (‘‘Minimum Design Loads for Buildings and Other Structures,’’ ASCE 7-98, American Society of Civil Engineers, 345 E. 47th St., New York, NY 10164-0619; ‘‘International Building Code 2000,’’ 1998.)
5.1.3 Factored Loads
Structural members must be designed with sufficient capacity to sustain without excessive deformation or failure those combinations of service loads that will pro- duce the most unfavorable effects. Also, the effects of such conditions as ponding of water on roofs, saturation of soils, settlement, and dimensional changes must be included. In determination of the structural capacity of a member or structure, a safety margin must be provided and the possibility of variations of material prop- erties from assumed design values and of inexactness of capacity calculations must be taken into account. Building codes may permit either of two methods, allowable-stress design or load–and–resistance factor design (also known as ultimate-strength design), to be used for a structural material. In both methods, design loads, which determine the required structural capacity, are calculated by multiplying combinations of service loads by factors. Different factors are applied to the various possible load combi- nations in accordance with the probability of occurrence of the loads. In allowable-stress design, required capacity is usually determined by the load combination that causes severe cracking or excessive deformation. For the purpose, dead, live, wind, seismic, snow, and other loads that may be imposed simultane- ously are added together, then multiplied by a factor equal to or less than 1. Load combinations usually considered in allowable-stress design are
(1) D L ( Lr or S or R )
(2) D L ( W or E / 1.4)
(3) D L W S / 2
(4) D L S W / 2
(5) D L S E / 1.
(6) 0.9 D E / 1.
where D dead load L live loads due to intended use of occupancy, including partitions Lr roof live loads S snow loads R rain loads W wind loads E seismic loads
STRUCTURAL THEORY 5.
Building codes usually permit a smaller factor when the probability is small that combinations of extreme loads, such as dead load plus maximum live load plus maximum wind or seismic forces, will occur. Generally, for example, a factor of 0.75 is applied to load-combination sums (2) to (6). Such factors are equivalent to permitting higher allowable unit stresses for the applicable loading conditions than for load combination (1). The allowable stress is obtained by dividing the unit stress causing excessive deformation or failure by a factor greater than 1. In load–and–resistance factor design, the various types of loads are each mul- tiplied by a load factor, the value of which is selected in accordance with the probability of occurrence of each type of load. The factored loads are then added to obtain the total load a member or system must sustain. A structural member is selected to provide a load-carrying capacity exceeding that sum. This capacity is determined by multiplying the ultimate-load capacity by a resistance factor, the value of which reflects the reliability of the estimate of capacity. Load criteria generally used are as follows:
1. 1.4 D 2. 1.2 D 1.6 L 0.5( Lr or S or R ) 3. 1.2 D 1.6( L (^) r or S or R ) (0.5 L or 0.8 W ) 4. 1.2 D 1.3 W 0.5 ( L (^) r or S or R ) 5. 1.2 D 1.0 E (0.5 L or 0.2 S ) 6. 0.9 D (1.3 W or 1.0 E )
For garages, places of public assembly, and areas for which live loads exceed 100 lb / ft^2 , the load factor usually is taken as unit for L in combinations 3, 4, and 5. For roof configurations that do not shed snow off the structure, the load factor should be taken as 0.7 for snow loads in combination 5. For concrete structures where load combinations do not include seismic forces, the factored load combinations of ACI 318 Section 9.2 shall be used. For both allowable stress design and strength design methods, elements and components shall be designed to resist the forces due to special seismic load com- binations
a) 1.2 D 0.5 L Em
b) 0.9 D Em
For floors in places of public assembly, for live load in excess of 100 psf, and for parking garage live load, the load factor is taken as 1.0 for L. Em is the maximum seismic effect of horizontal and vertical forces.
5.2 STRESS AND STRAIN
Structural capacity, or ultimate strength, is that property of a structural member that serves as a measure of is ability to support all potential loads without severe crack- ing or excessive deformations. To indicate when the limit on load-carrying useful- ness has been reached, design specifications for the various structural materials establish allowable unit stresses or design strengths that may not be exceeded under
STRUCTURAL THEORY 5.
Suppose, for the truss in Fig. 5.1, the reactions at the supports are to be com- puted. Taking moments about the right end and equating to zero yields 40 R (^) l 30 20 0, from which left reaction R (^) L 600 / 40 15 kips. Equating the sum of the vertical forces to zero gives 20 15 RR 0, from which the right reaction RR 5 kips.
5.2.2 Unit Stress and Strain
To ascertain whether a structural member has adequate load-carrying capacity, the designer generally has to compute the maximum unit stress produced by design loads in the member for each type of internal force—tensile, compressive, or shear- ing—and compare it with the corresponding allowable unit stress. When the loading is such that the unit stress is constant over a section under consideration, the stress may be obtained by dividing the force by the area of the section. But in general, the unit stress varies from point to point. In that case, the unit stress at any point in the section is the limiting value of the ratio of the internal force on any small area to that area, as the area is taken smaller and smaller. Sometimes in the design of a structure, unit stress may not be the prime con- sideration. The designer may be more interested in limiting the deformation or strain. Deformation in any direction is the total change in the dimension of a member in that direction. Unit strain in any direction is the deformation per unit of length in that direc- tion. When the loading is such that the unit strain is constant over a portion of a member, it may be obtained by dividing the deformation by the original length of that portion. In general, however, the unit strain varies from point to point in a member. Like a varying unit stress, it represents the limiting value of a ratio.
5.2.3 Hooke’s Law
For many materials, unit strain is proportional to unit stress, until a certain stress, the proportional limit, is exceeded. Known as Hooke’s law, this relationship may be written as
ƒ ƒ E or (5.20) E
where ƒ unit stress unit strain E modulus of elasticity
Hence, when the unit stress and modulus of elasticity of a material are known, the unit strain can be computed. Conversely, when the unit strain has been found, the unit stress can be calculated. When a member is loaded and the unit stress does ot exceed the proportional limit, the member will return to its original dimensions when the load is removed. The elastic limit is the largest unit stress that can be developed without a permanent deformation remaining after removal of the load. Some materials possess one or two yield points. These are unit stresses in the region of which there appears to be an increase in strain with no increase or a small
5.20 SECTION FIVE
FIGURE 5.5 Bracket in shear. FIGURE 5.6 Bearing load and pressure.
FIGURE 5.3 Tension member. FIGURE 5.4 Compression member.
decrease in stress. Thus, the materials exhibit plastic deformation. For materials that do not have a well-defined yield point, the offset yield strength is used as a measure of the beginning of plastic deformation. The offset yield strength, or proof stress as it is sometimes referred to, is defined as the unit stress corresponding to a permanent deformation, usually 0.01% (0.0001 in / in) or 0.20% (0.002 in / in).
5.2.4 Constant Unit Stress
The simplest cases of stress and strain are those in which the unit stress and strain are constant. Stresses due to an axial tension or compression load or a centrally applied shearing force are examples; also an evenly applied bearing load. These loading conditions are illustrated in Figs. 5.3 to 5.6. For the axial tension and compression loadings, we take a section normal to the centroidal axis (and to the applied forces). For the shearing load, the section is taken along a plane of sliding. And for the bearing load, it is chosen through the plane of contact between the two members.
