5.Loads

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


Seismic disturbance

Live load


Wind load (gusts)

Temperature load (shrinkage?)


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 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


小震不坏中震可修大震不倒

Earthquake resistance calls for energy absorption (or ductility) rather than strength only


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


2. The dynamic properties of the building, such as its mode shapes and periods of vibration and its damping characteristics


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


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


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


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


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


3m 3m 3m

5m


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


3m 3m 3m

5m


Alternatively, UDL on beam can be calculated as= 7.1 × 3 + 0.5= 21.8 kN/m

Loading


3m 3m 3m

5m


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


6m RB1 RC1

Beam B1-C1


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



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



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



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



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

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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 *



Deflection

Load

Elas

tic ra

nge

of lo

ad

Plas

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rang

e


Ultimate load

Response of structures

Dead load

Live load

Wind or EQ load *



Deflection

Load

Elas

tic ra

nge

of lo

ad

Plas

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e


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 *



Deflection

Load

Elas

tic ra

nge

of lo

ad

Plas

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rang

e


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 *



Deflection

Load

Elas

tic ra

nge

of lo

ad

Plas

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rang

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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

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5.Loads