5.loads theory and design of structures 1 university of hongkong
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University of HongKongTRANSCRIPT
Theory and Design of Structures I
Structural Loads & Response
IntroductionAll structures are composed of a number of interconnected elements. They enable the internal/external loads to be safely transmitted down to the ground, e.g.– slabs– beams– columns– walls – foundations
Sequence of load transfer
Reactions
“Loads”
R1 R2
R1 R2
It is usually assumed that the reaction from one element is a load on the next
Sequence of load transferFrom roof slab to beam
From floor slab to beam
From beam to column
From beam to column
From column to foundation
Transfer of loading
Sequence of load transfer – load path
River
Tributary
Sequence of load transfer
Sequence of load transfer is not clear.
The design processThe designer must make an assessment of the future likely level of loading to which the structure may be subjected during its design life.
Determination of design loads acting on the structure
Determination of design loads on individual elements
Calculation of bending moments, shear forces and deflections of beams
Sizing of beams
Sizing of columns
Nature of loading & design loads
Nature of loading & design loads
Seismic disturbance
Live load
Nature of loading & design loads
Wind load (gusts)
Temperature load (shrinkage?)
Nature of loading & design loads
Foundation settlement
Impact
Nature of loading & design loadsIt is usually assumed that the dynamic loads on the building structures can be reduced to equivalent static loads, e.g.– live loads– seismic disturbances– gusting of wind– movement of machinery
Actual loading: dynamic (not static); changingFor design: equivalent static load
LL uniform design load (on buildings)basic LL + impact allowance (on bridges)
WL equivalent static load(kN/m2 of exposed surface area)
EQL equivalent static load(% of gravity load)
Others: essentially STATIC
Nature of loading & design loads
Nature of loading & design loadsThe loads acting on a structure are divided into different basic types:– dead load – live load– wind load– earthquake load– loading from other sources
For each type, the characteristic and design values must be estimated
The designer will have to determine the particular combination of loading which is likely to produce the most adverse effect on the structure in terms of bending moments, shear forces, deflections, etc.
Nature of loading & design loadsLive load Wind
load
Max axial load
Combination of loading
Most adverse effects
Dead Loads (DL)
Dead Loads (DL)DLs are all the permanent loads acting on the structure including:– self-weight– finishes– fixtures– partitions
Dead Loads (DL)Estimation of the self-weight of an element – cyclic process since its value can only be
assessed once the element has been designed
LL DL
? ?
Dead Loads (DL)Assume a cross-section
DL
BM, SF, etc
Check if OK
Revise cross-section
Yes No
End
LL
Economical?
Imposed Loads (IL) / Live Loads (LL)
Live Loads (LL)Imposed load or live load represents the load due to the proposed occupancy and includes:– the weights of the occupants and furniture– roof loads including snow
They are much more variable than DL, and are more difficult to predict
Live Loads (LL)Heavy live loads are rare
There are a few medium live loads
Most of the live loads are light
Live Loads (LL)It is possible to concentrate a heavy load over a rather small area (0.2-0.6 m2) amounting to, say, 25 or 50 kN/m2 on that small area.
Bigger equivalent UDL!
When a large tributary area (over 10 or 15 m2) is supported by a primary structural component, the significance of that concentration as compared with the overall load will be reduced correspondingly
Live Loads (LL)
Smaller equivalent UDL
An average design load value can be assigned when the actual or probable type of building occupancy is known– basic live load for application when considering
the larger tributary areasFor smaller areas, the effect of concentrated live load should be considered as a special case
Live Loads (LL)Loading on a one-way slab supported on four beams
Live Loads (LL)
Column
Primary beam
Secondary beam
Live Loads (LL)
Large tributary area of a primary beam
Small tributary area of a secondary beam
Medium tributary area of a corner column
Different types of live loads:L.L. on floors (depending on uses)L.L. on roofL.L. on bridges (impact factor or formula)
Live Loads (LL)
Wind Loads (WL)
Wind Loads (WL)Wind load on a building is dynamic, but it is conveniently expressed as equivalent static load in kN/m2 of exposed surface area
Wind loads vary with wind speed, surface shape, exposed area, etc
Wind pressure can either add to the other gravitational forces acting on the structure or, equally well, exert suction or negative pressures on the structure
Wind Loads (WL)
Suction or negative pressure
Positive pressure
Wind
Wind pressure primarily depends on– its velocity– the slope and shape of the surface– the protection from wind offered by other
structuresand to a smaller degree– the density of the air– the surface texture
Wind Loads (WL)
Examples of wind-sensitive structures:– long-span bridges (suspension bridges and
cable-stayed bridges)– tall buildings– slender towers
Wind tunnel tests are often needed
Wind Loads (WL)
The Structural Engineer – 15 November 2005
Earthquake Loads (EQL)
Earthquake Loads (EQL)EQL– mainly lateral loads produced by earthquake– dynamic, expressed as % of overall mass or
gravity load (W) of a building
The % may vary from – 2% to 5% (of W) for tall buildings in moderate
seismic zones – 10% to 20% for short stiff buildings in active
seismic zones
There are two basic objectives in design for earthquake:
1. To protect the public from loss of life and serious injury and to prevent buildings from collapse and dangerous damage under a maximum-intensity earthquake
2. To ensure buildings against any but very minor damage under moderate to heavy earthquakes
Earthquake Loads (EQL)
小震不坏 中震可修 大震不倒
Earthquake resistance calls for energy absorption (or ductility) rather than strength only
Earthquake Loads (EQL)
P
∆ Ductile Brittle
P
∆
No good! Desirable!
Actual seismic loads depend on the following factors:
1. The intensity and character of the ground motion as determined at the source and its transmission to the building, e.g.
– max. ground acceleration,– frequency spectrum,– direction of motion, etc
Earthquake Loads (EQL)
2. The dynamic properties of the building, such as its mode shapes and periods of vibration and its damping characteristics
Earthquake Loads (EQL)
f0
≅
f1 f2
Frequencies
Mode shapes
…
…
Effect of damping
t
∆
3. The mass of the building as a whole or of its components– Mass of building– F = m a– Certain amount of overstress allowed
Earthquake Loads (EQL)
Internal & External Movements in Structures
Int. & ext. movement in structuresInternal movements or strains in a structure can be produced as a result of differential movement due to temperature variationacross the structure.
Int. & ext. movement in structuresIf a structure is entirely free to expand and contract under temperature changes, then there may be no internal stresses produced.
Uniform rise in temperature
Linear distribution of temperature
No stress induced
Different parts of a building will be exposed to, and will respond differently to, environmental conditions Hot
Stresses induced
Hot Cold
Cold
Hot
HotStresses induced
Int. & ext. movement in structures
To minimize the internal stresses and strains, provisions of expansion joints (or movement joints) is necessary, particularly along the roof lines and the outside walls of a buildingSuch provisions may be unsightly and expensive
Int. & ext. movement in structures
Mov
emen
t joi
nt
Elevation of a large building
Movement joint (MJ)
Bearing Bearing
Movement joint
Abutment Abutment
Certain material such as concrete tends to shrink and/or creep under load as time goes on, and hence produces differential strains of one floor versus the other. Such deformations may also create stresses.Forces may also be created by unequal settlement of the foundations– uniform settlement, no serious forces– uneven foundation settlement may result in
undesirable stresses and strains
Int. & ext. movement in structures
Examples
Calculate the self-weight of a reinforced concrete beam of breadth 300 mm, depth 600 mm and length 6000 mm.
Assuming that unit mass of reinforced concrete is 2400 kg/m3 and the gravitational constant is 10 m/s2 (strictly 9.807 m/s2), the unit weight of reinforced concrete, ρ, is
ρ = 2400 × 10 = 24 000 N/m3 = 24 kN/m3
Hence, the self-weight of beam, SW, isSW= volume × unit weight= (0.3 × 0.6 × 6) × 24 = 25.92 kN
Example 1 Self-weight of a reinforced concrete beam
A composite floor consisting of a 150 mm thick RC slab supported on steel beams spanning 5 m and spaced at 3 m centres is to be designed to carry an imposed load of 3.5 kN/m2. Assuming that the unit mass of the steel beams is 50 kg/m run, calculate the design loads on a typical internal beam.
Example 2 Design loads on a floor beam
3m 3m 3m
5m
Example 2: Design loads on a floor beam. RC (ρ = 2400kg/m3, gravitational constant 10m/s2)– 2400 × 10 = 24 000 N/m3 = 24 kN/m3
Steel beams – Unit mass of beam = 50 kg/m run– Unit weight of beam
= 50 × 10 = 500 N/m run = 0.5 kN/m run
Unit weights of materials
Example 2 Design loads on a floor beam
3m 3m 3m
5m
Example 2: Design loads on a floor beam.
Slab– DL = 0.15 × 24 = 3.6 kN/m2
– IL= 3.5 kN/m2
– Total load = 3.6 + 3.5 = 7.1 kN/m2
Beam– DL = 0.5 kN/m run
Loading
Example 2 Design loads on a floor beam
3m 3m 3m
5m
Example 2: Design loads on a floor beam.
Total load (each internal beam supports a uniformly distributed load from a 3 m width of slab plus self-weight)Design load on beam = slab load + self-weight of beam
= 7.1 × 5 × 3 + 0.5 × 5 = 109 kNUDL on beam = 109 kN / 5 m = 21.8 kN/m
Loading
Example 2 Design loads on a floor beam
3m 3m 3m
5m
Example 2: Design loads on a floor beam.
Alternatively, UDL on beam can be calculated as= 7.1 × 3 + 0.5= 21.8 kN/m
Loading
Example 2 Design loads on a floor beam
3m 3m 3m
5m
Example 2: Design loads on a floor beam.
Example 3 Design loads on floor beams and columnsThe floor shown below with an overall depth of 225 mm is to be designed to carry an imposed load of 3 kN/m2 plus floor finishes and ceiling loads of 1 kN/m2. Calculate the design loads acting on beams B1-C1, B2-C2 and B1-B3 and columns B1 and Cl. Assume that all the column heights are 3 m and that the beam and column weights are 70 and 60 kg/m run respectively.
6m
3m
3m
1
A C
2
3
3m
B
Example 3. Design loads on floor beams and columns.
Example 3 Design loads on floor beams and columns
RC (ρ = 2400kg/m3, gravitational constant 10m/s2)– 2400 × 10 = 24 000 N/m3 = 24 kN/m3
Steel beams – Unit mass of beam = 70 kg/m run– Unit weight of beam
= 70 × 10 = 700 N/m run = 0.7 kN/m runSteel columns – Unit mass of column = 60 kg/m run– Unit weight of column
= 60 × 10 = 600 N/m run = 0.6 kN/m run
Unit weights of materialsExample 3 Design loads on floor beams and columns
Slab– DL (SW) = 0.225 × 24 = 5.4 kN/m2
– DL (FF) = 1 kN/m2
– Total DL = 5.4 + 1 = 6.4 kN/m2
– IL= 3 kN/m2
– Total load = 6.4 + 3 = 9.4 kN/m2
Beam– DL = 0.7 kN/m run
Column– DL = 0.6 kN/m run
Loading
Design load on beam B1-C1= slab load + self-weight of beam= 9.4 × 6 × 1.5 + 0.7 × 6= 88.8 kNRB1 = RC1 = 88.8 / 2 = 44.4 kN
Beam B1-C1
6m
3m
3m
1
A C
2
3
3m
B
Example 3. Design loads on floor beams and columns.
6m RB1 RC1
Beam B1-C1
Example 3 Design loads on floor beams and columns
Design load on beam B2-C2= slab load + self-weight of beam= 9.4 × 6 × 3 + 0.7 × 6= 173.4 kNRB2 = RC2 = 173.4/2 = 86.7 kN
Beam B2-C2
6m
3m
3m
1
A C
2
3
3m
B
Example 3. Design loads on floor beams and columns.
Example 3 Design loads on floor beams and columns
6mRB2 RC2
Beam B2-C2
Design load on beam B1-B3= slab load + self-weight of beam + point load RB2
= (9.4 × 1.5 × 6 + 0.7 × 6) + 86.7= 88.8 + 86.7 = 175.5 kNRB1 = RB3 = 175.5/2 = 87.75 kN
Beam B1-B3
6m
3m
3m
1
A C
2
3
3m
B
Example 3. Design loads on floor beams and columns.
Example 3 Design loads on floor beams and columns
3m RB1 RB3
Beam B1-B3
3m
Beam B1-C1: RB1 = 44.4 kNBeam B1-B3: RB1 = 87.75 kNBeam A1-B1: RB1 = 0.7×3 / 2 = 1.05 kN (self-wt only)Column B1 = 0.6×3 = 1.8 kN (self-wt only)Total load = = 44.4 + 87.75 + 1.05 + 1.8 = 135 kN
Column B1
6m
3m
3m
1
A C
2
3
3m
B
Example 3. Design loads on floor beams and columns.
Example 3 Design loads on floor beams and columns
Column B1
Beam A1-B1 Beam B1-C1
Beam
B1-
B3
Beam B1-C1: RC1 = 44.4 kNBeam C1-C3: RC1 = (86.7 + 4.2)/2 = 45.45 kNColumn C1 = 0.6×3 = 1.8 kN (self-wt only)Total load = = 44.4 + 45.45 + 1.8 = 91.65 kN
Column C1
6m
3m
3m
1
A C
2
3
3m
B
Example 3. Design loads on floor beams and columns.
Example 3 Design loads on floor beams and columns
Column C1
Beam B1-C1
Beam
C1-
C3
Response of Structures
Response of structures The structure
must be able to respond with proper behaviour and prescribed stability
Dead load
Live load
Wind or EQ load *
Elastic behaviour Plastic behaviour
Reserve load capacity
Deflection
Load
Elas
tic ra
nge
of lo
ad
Plas
tic
rang
e
Life history of a structure (* only partial or zero live load is considered together with wind or EQ load).
Ultimate load
Response of structures
DL only– Very little deflection,
if any, in the lateral direction
LL + DL– More deflection and
higher stresses are produced locally
Dead load
Live load
Wind or EQ load *
Elastic behaviour Plastic behaviour
Reserve load capacity
Deflection
Load
Elas
tic ra
nge
of lo
ad
Plas
tic
rang
e
Life history of a structure (* only partial or zero live load is considered together with wind or EQ load).
Ultimate load
Response of structures
Dead load
Live load
Wind or EQ load *
Elastic behaviour Plastic behaviour
Reserve load capacity
Deflection
Load
Elas
tic ra
nge
of lo
ad
Plas
tic
rang
e
Life history of a structure (* only partial or zero live load is considered together with wind or EQ load).
Ultimate load
Response of structuresWL or EQL– higher forces and
stresses are produced in various components
– one-third or so increase in allowable stresses is permitted since these loads occur rather infrequently
Dead load
Live load
Wind or EQ load *
Elastic behaviour Plastic behaviour
Reserve load capacity
Deflection
Load
Elas
tic ra
nge
of lo
ad
Plas
tic
rang
e
Life history of a structure (* only partial or zero live load is considered together with wind or EQ load).
Ultimate load
Response of structuresReserve load capacity– takes care of
unexpected events, e.g. high wind (margin of safety)
– keeps the behaviour of the structure within tolerable limits of movement and strain under the normally expected high wind or earthquake condition
Dead load
Live load
Wind or EQ load *
Elastic behaviour Plastic behaviour
Reserve load capacity
Deflection
Load
Elas
tic ra
nge
of lo
ad
Plas
tic
rang
e
Life history of a structure (* only partial or zero live load is considered together with wind or EQ load).
Ultimate load
Response of structuresUnder catastrophic earthquakes, the building is permitted to extend into plastic range so that certain portions of the building will suffer minor damage
Building Codes, Structural Behaviour and Strength
Building codesIt is normal practice to design buildings according to building code requirementsCodes set up minimum requirements and serve as rough guides for design. They specify:– minimum loading to be considered– maximum stresses not to be exceeded
Specified loading and allowable stresses are used as empirical approximations
“Allowable stress” or “permissible stress”approach
σc ≤ fc / (FOS)
σt ≤ ft / (FOS)
P
Structural behaviour and strength Structural behaviour and strengthLoad factor method or ultimate strength method is a more rational approach than the allowable stress approach– the specified load has to be multiplied by a
factor to be equated to the (least) ultimate strength of the structure
safetyofFactorSafe uPP =
Pu
Structural behaviour and strengthOther points to check:– deflections– vibrations– cracks– human sensitivity to vibration* Simple rules often used (e.g. span / depth ratio,
aspect ratio, etc)
Drawback of load factor method– difficult to predict the actual load capacity of a
building– uncertain about whether the structure behaves
properly, e.g. excessive deflections/vibrations, cracks, etc
Limit state design method is used instead in which both ultimate limit state and serviceability limit state are addressed
Structural behaviour and strength
The End