Theory and Design of Structures I

Structural Loads & Responses

 Introduction
All 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
From roof slab to beam

From beam Transfer of loading

to column

From floor slab to beam

From beam
to column

From column
to foundation

It is usually assumed that the reaction from

one element is a load on the next
Sequence of load transfer – load path

Tributary

River
 The design process
The 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
The 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
Nature of loading & design loads

Live load Seismic

disturbance

Slow variations Rapid variations

Nature of loading & design loads
Rapid variations Slow variations

Wind load Temperature load

(gusts) (shrinkage?)
Nature of loading & design loads

Foundation
settlement Impact
Slow variations Rapid variations
 Nature of loading & design loads
It is usually assumed that the dynamic loads
on the building structures can be reduced to
equivalent static loads, e.g.
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
Wind Live load
load Combination
of loading

Max
axial Most adverse
load effects

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.
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
Revise
LL BM, SF, etc cross-
section

Check if OK
No
Yes
End 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!

 Live Loads (LL)
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

Smaller equivalent UDL

 Live Loads (LL)
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 areas
For smaller areas, the effect of concentrated live
load should be considered as a special case
Live Loads (LL)
Live Loads (LL)
 Live Loads (LL)
Loading on a one-way slab supported on four
beams (approximate model)
 Live Loads (LL)
Column
Primary beam Medium tributary
area of a corner
Secondary beam column

Large tributary area Small tributary area

of a primary beam of a secondary beam
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 primarily depends on

– its velocity
– the slope and shape of the surface
– the protection from wind offered by other structures
 Wind Loads (WL)
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
Suction or
Positive
negative
pressure pressure
 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

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

P P

 
Brittle Ductile

No good! Desirable!
Internal & External Movements
in Structures
Int. & ext. movement in structures
Internal movements or strains in a structure
can be produced as a result of differential
movement due to temperature variation
across the structure.
Other sources:
- shrinkage
- foundation settlement
Int. & ext. movement in structures
If 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
Int. & ext. movement in structures
Different parts of a building will be exposed to,
and will respond differently to, environmental
conditions Hot

Stresses induced

Hot

Cold Stresses induced

Hot Hot
Cold
 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 building
Such provisions may be unsightly and expensive
Movement Movement
joint (MJ) joint

Abutment Bearing Bearing Abutment

Movement joint
MJ Elevation of a large building
Bearing
Response of Structures
Response of structures
 Response of structures
The structure Elastic behaviour Plastic behaviour Ultimate
must be able to load

respond with

Plastic
range
Reserve load
proper behaviour capacity
and prescribed

Load
stability

Elastic range of load

Wind or EQ load *

Live load

Dead load

Deflection
Life history of a structure (* only partial or zero live load is
considered together with wind or EQ load).
 Response of structures
DL only Elastic behaviour Plastic behaviour Ultimate
load
– Very little deflection,

Plastic
range
Reserve load
if any, in the lateral capacity
direction

Load
LL + DL

Elastic range of load

Wind or EQ load *

– More deflection and Live load

higher stresses are

produced locally Dead load

Deflection
Life history of a structure (* only partial or zero live load is
considered together with wind or EQ load).
 Response of structures
WL or EQL Elastic behaviour Plastic behaviour Ultimate
load
– higher forces and

Plastic
range
stresses are produced Reserve load
capacity
in various

Load
components

Elastic range of load

Wind or EQ load *
– one-third or so Live load
increase in allowable
stresses is permitted Dead load

since these loads

occur rather Deflection
Life history of a structure (* only partial or zero live load is
infrequently considered together with wind or EQ load).
 Response of structures
Reserve load capacity Elastic behaviour Plastic behaviour Ultimate
load
– takes care of

Plastic
range
unexpected events, e.g. Reserve load
capacity

high wind (margin of

Load
safety)

Elastic range of load

Wind or EQ load *

– keeps the behaviour of Live load

the structure within

tolerable limits of Dead load

movement and strain

Deflection
under the normally Life history of a structure (* only partial or zero live load is
expected high wind or considered together with wind or EQ load).

earthquake condition
 Response of structures
Under catastrophic Elastic behaviour Plastic behaviour Ultimate

earthquakes, the
load

Plastic
range
building is permitted Reserve load
capacity

to extend into plastic

Load
range so that certain

Elastic range of load

Wind or EQ load *

portions of the Live load

building will suffer

Dead load
minor damage
Deflection
Life history of a structure (* only partial or zero live load is
considered together with wind or EQ load).
 Examples
Notes:
1. Only nominal loads are calculated.
2. There is no need to apply partial safety factors.
Example 1 Self-weight of a reinforced concrete beam

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, , is
 = 2400  10 = 24 000 N/m3 = 24 kN/m3
Hence, the self-weight of beam, SW, is
SW= volume  unit weight
= (0.3  0.6  6)  24 = 25.92 kN
 Example 2 Design loads on a floor beam

5m
3m 3m 3m

Example 2: Design loads on a floor 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

5m
3m 3m 3m

Unit weights of materials 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
per m or
– Unit mass of beam = 50 kg/m run per m run
– Unit weight of beam
= 50  10 = 500 N/m run = 0.5 kN/m run
 Example 2 Design loads on a floor beam

5m
Loading 3m 3m 3m

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
 Example 2 Design loads on a floor beam

5m
3m 3m 3m
Loading 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 kN
UDL on beam = 109 kN / 5 m = 21.8 kN/m
 Example 2 Design loads on a floor beam

5m
3m 3m 3m
Loading 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
Example 3 Design loads on floor beams and columns
The 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.
3

3m

3m

1
3m 6m

A B C
Example 3. Design loads on floor beams and columns.
Example 3 Design loads on floor beams and columns
Unit weights of materials

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 run
Steel columns
– Unit mass of column = 60 kg/m run
– Unit weight of column
= 60  10 = 600 N/m run = 0.6 kN/m run
Example 3 Design loads on floor beams and columns
Loading
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
Example 3 Design loads on floor beams and columns
3

RB1 RC1 3m
6m

Beam B1-C1 2

3m

1
3m 6m

A B C
Example 3. Design loads on floor beams and columns.
Beam B1-C1
Design load on beam B1-C1
= slab load + self-weight of beam
= 9.4  6  1.5 + 0.7  6
= 88.8 kN
RB1 = RC1 = 88.8 / 2 = 44.4 kN
Example 3 Design loads on floor beams and columns
3

RB2 RC2 3m
6m

2
Beam B2-C2
3m

1
3m 6m

A B C
Example 3. Design loads on floor beams and columns.
Beam B2-C2
Design load on beam B2-C2
= slab load + self-weight of beam
= 9.4  6  3 + 0.7  6
= 173.4 kN
RB2 = RC2 = 173.4/2 = 86.7 kN
Example 3 Design loads on floor beams and columns
3

3m

RB1 RB3
3m 3m 2

Beam B1-B3 3m

1
3m 6m

A B C
Beam B1-B3 Example 3. Design loads on floor beams and columns.

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 kN
RB1 = RB3 = 175.5/2 = 87.75 kN
Example 3 Design loads on floor beams and columns

Beam B1-B3
3

3m

Beam A1-B1 Beam B1-C1 2

Column B1 3m

1
3m 6m

A B C
Column B1 Example 3. Design loads on floor beams and columns.

Beam B1-C1: RB1 = 44.4 kN

Beam B1-B3: RB1 = 87.75 kN
Beam 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
Example 3 Design loads on floor beams and columns
3

Beam C1-C3
3m

2
Beam B1-C1
3m
Column C1
1
3m 6m

A B C
Column C1 Example 3. Design loads on floor beams and columns.

Beam B1-C1: RC1 = 44.4 kN

Beam C1-C3: RC1 = (86.7 + 4.2)/2 = 45.45 kN
Column C1 = 0.63 = 1.8 kN (self-wt only)
Total load = = 44.4 + 45.45 + 1.8 = 91.65 kN
 Example 4: Floor beams for an office building
The steel beams for part of the floor of a library with book storage
are shown in Figure 13. The floor comprises an RC slab supported
on universal beams. The design loading has been estimated as:
Dead load – slab, self-weight of steel, finishes, ceiling, partitions,
services and fire protection: = 6 kN/m2
Imposed load: = 4 kN/m2
Determine the loadings for beams A2-B2 and B1-B3.
3

3m
2
3m

5m 5m

A B C
Figure 13. Part floor plan of a library 64
 Beam A2-B2
The aspect ratio of each slab is 5/3 = 1.67 < 2. It is
taken as a two-way slab. The tributary areas are shown
in the diagram.
Beam A2-B2

Note: Slabs with aspect 3

ratio between 1.5 and 2
are sometimes taken as
3m
one-way slab although it
2
results in more error.
3m

5m 5m

A B C
Figure 13. Part floor plan of a library 65
 Beam A2-B2
1. Draw diagrams of beam A2-B2 showing the applied
loading.
Service dead load  6  1.5  2  18 kN/m
Service imposed load  4  1.5  2  12 kN/m
12kN/m
3

3m
Imposed load
21 21 2

3m
18kN/m
1

5m 5m

A B C
1.5 m 2.0 m 1.5 m Figure 13. Part floor plan of a library

31.5 Dead load 31.5

66
Figure 14. Working Loads on Beam A2-B2 (kN)
 Beam B1-B3
1. Draw diagrams of beam A2-B2 showing the applied
loading. The resulting triangular loads are
DL:  2  (3  1.5 / 2)  6  27 kN
IL:  2  (3  1.5 / 2)  4  18 kN
12kN/m 42 12kN/m Beam B1-B3

Imposed load
39 39 3

18kN/m 63 18kN/m

3m
2
3m 3m

3m
58.5 58.5
Dead load
1
Figure 16. Working loads on beam B1-B3 (kN)
5m 5
A B
67
Figure 13.Partfloorplanof a lib
The End