Chapter 3 - Loads

ENGR 327 Reinforced Concrete Design I

Dr. Solomon Tesfamariam

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CHAPTER 3 BUILDING AND DESIGN CODES Two broad classifications of codes General building codes Deign codes

3.1 BUILDING CODES AND DESIGN STANDARDS National Building Code of Canada (NBC 2005)

regulate design and construction of buildings provides design loads refers to appropriate design standards

Design of Concrete Structures (CSA A23.3-04)

covers concrete design based on limit states design

3.1.1 DESIGN LOADS The design and construction of buildings in Canada is regulated by the National Building Code of Canada (NBC). Building Codes provide guidance with expected loads on structures

(NBC 4.1.2.1) i. deal load (D) ii. live load due to use and occupancy (L) iii. snow load (S) iv. wind load (W) v. earthquake load (E) vi. effects of temperature change (T) vii. permanent load due to lateral earth pressure (H) viii. permanent effect caused by prestress (P) These loads can be grouped in the predominant load direction. In general, the loads can broadly be classified as gravity or lateral loads. The gravity loads entails of dead and live loads. Whereas, the lateral loads entails wind and earthquake loads.

Load Classification (Brzev and Pao 2006)

Loads

Gravity

Dead Live Snow

Lateral

Wind Earthquake



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3.1.2 LOAD PATHS IN STRUCTURES • Most structural components can be categorized as beams, columns, slabs

and walls • These components have one or two dimensions considerably smaller than

the other(s) • It is convenient to integrated stresses through the smaller dimension(s)

and talk in terms of stress resultants (axial forces, shear forces and bending moments)

• A component (or member) carries load applied in the direction of the smaller dimension through shear force (V) and bending moment (M)

• A component carries load applied normal to the direction of the smaller dimension through axial force (N)

Illustrations of load paths in typical concrete structures

Load paths in a frame subjected to gravity loads

Load paths in a frame subjected to lateral loads



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3.2 TRIBUTARY AREAS FOR FLOOR AND ROOF LOADS Tributary area is related to the load path.

The concept is used to determine the loads on beams and columns due to uniform floor and roofs loads.

a) One way slab b) Two way slab

Defined by the lines of support of the member, and the lines of zero shear in

the members supported.

• For simple span construction, lines of zero shear are normally located near mid span.

• For continuous construction, a structural analysis may be required to determine lines of zero shear.



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3.2.1 TRIBUTARY AREA FOR COLUMNS Column tributary area is the area surrounding the column that is bounded by

the floor centerlines Column loads may also be based on beam end reactions.



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3.2.2 TRIBUTARY AREA FOR BEAMS Area of slab supported by a particular beam Area define by floor centerline and/ or 45o line (two-way action)

One-Way Action

Two-way action

Combined



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See also: Commentary F “Tributary Area” – NBC 2005 Structural Commentaries

Tributary area for a two-way slab with joists, beams and girders

Tributary area for flat slab without beams and girders

Tributary area for a two-way slab with beam

Tributary area for a one-way deck or slab with joists, beam and girders

Tributary area for a one-way slab with girders



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Example A partial floor plan of a reinforced concrete building is shown in the figure below. The roof is subjected to a total uniform area load (w) of 15.0 kPa (including the floor self-weight). Determine the uniform load on the typical beam B1 and the typical interior girder G1 in the figure.



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3.3 DEAD LOADS

Dead loads include the weight of the structure and any permanent attachments such as fixed partitions, ceiling, floor and roof coverings, permanent equipments, etc. • Self-weight • Superimposed dead load

Dead load may include loads due to earth, plants and trees (specific recognition in NBC 2005).

Dead loads are generally well defined and can be predicted reasonably well in advance lower load factor

Erection or construction dead loads may also be present may change during construction stages

Typical unit weights for construction materials are listed below See NBC 2005 Subsection 4.1.4.

Unit Weights for Construction Materials

Material Unit weight

Concrete Low density Semi-low density Plain Reinforced

18.1 kN/m

3

21.0 kN/m3

23.5 kN/m3

24.0 kN/m3

Steel 77.0 kN/m3

Wood Hardwood softwood

9.5 kN/m

3

6.0 kN/m3

Aluminum 27.0 kN/m3



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3.4 LIVE LOAD DUE TO USE AND OCCUPANCY

Live load: any non-permanent load on the structure produced by building occupants, furniture storage items, minor equipment, movable partitions, etc. defined according to the “occupancy and use” of the building.

Live load is relatively short term and possibly variable in magnitude and position not easy to predict or quantify.

Load factors for live loads are normally higher than those for dead loads.

3.4.1 VARIABILITY IN MAGNITUDE

To deal with the uncertainty of live load magnitude, building codes have enacted conservative values for specified minimum live loads based on experience and judgement rather than from systematic surveys of loading.

3.4.2 VARIABILITY IN POSITION

Because live loads are transient, pattern loading must be considered. Maximum force effects in structural members may be produced by

patterns of live loading rather than live load applied on all spans (see Figure).

Pattern Loading for Maximum Load Effect in Column AB

3.4.3 NBC 2005 LIVE LOAD DUE TO USE AND OCCUPANCY

Provisions specified in Article 4.1.5. • Uniformly distributed loads (Table 4.1.5.3) • Concentrated loads (Table 4.1.5.10)

Where occupancy may vary at different times, the greatest value must be used.

Pattern loading must be considered



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For use/occupancy not specified in Table 4.1.5.3, live load is determined from an analysis of the weight from: • Probable assembly of persons • Probable accumulation of equipment/ furnishing • Probable storage of materials



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Concentrated loads are applied on an area of 750 mm by 750 mm located so as to cause maximum effects NBC 2005 requires that design is undertaken by considering the most unfavourable load combination from the following table.



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3.4.4 VARIATION OF LIVE LOAD WITH TRIBUTARY AREA

Also know as Live Load Reduction Building codes permit a reduced live load for individual structural members

based on the size of the tributary area contributing loads to the member. • Reflects probability that is unlikely that a member supporting a large

floor area will be subjected to full live loading over the entire area. NBC 2005 Article 4.1.5.9 specifies three condition for live load reduction:

Case Occupancy and

Design Live Load Member

Tributary Area Live Load

Reduction Factor

1) Assembly occupancies with live load less than 4.8 kPa

Any

1

2)

Assembly occupancies with live load of 4.8 kPa or more or Storage, manufacturing, retail stores, garages or footbridge

> 80 m2 0.5+√ ⁄

3) Any use or occupancy other than (1) and (2)

> 20 m2 0.3+√ ⁄

Note: = tributary area in square meters for type of use and occupancy, excluding the

area supporting snow

For design, the live load due to use and occupancy is the load provided in Article 4.1.5.3., multiplied by the reduction factor given above.

For columns in multi-storey buildings, the tributary area is the sum of the

tributary areas for the column at each floor level above the storey in question.



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Example A six-storey reinforced concrete office building is shown. Determine the dead load (DL) and live load (LL) carried by a typical interior column at each storey level. All columns are 400 x 400 mm. The slab is 200 mm thick. Design information: Super-imposed floor DL = 1.65 kPa

= 24 kN/m3



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Dead Load Calculations Table

Level Height

(m)

Dead load (kN)

Per level Cumulative

Roof 21 234.0 234.0

6 17.5 327.1 561.1

5 14 327.1 888.2

4 10.5 327.1 1215.3

3 7 327.1 1542.4

2 3.5 327.1 1869.5

Ground 0 13.4 1882.9



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Live Load Calculations Table

Level

Live Load Due to Occupancy

Live Load (kN) Tributary area (m2) LL Reduction

Factor Reduced LL

(kN) Per level Cumulative Per level Cumulative

Roof 0 0 0 0

6 234 234 48.75 48.75

5 234 468 48.75 97.50

4 234 702 48.75 146.25

3 234 936 48.75 195

2 234 1170 48.75 243.75

Ground 0 1170 48.75 243.75



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3.5 LOADS DUE TO SNOW AND RAIN

Independent of the building occupancy, but highly dependent on location. Must consider conditions that cause drifts and other concentrated

accumulations on the structure. Specified in Subsection 4.1.6 in the NBC 2005. Additional information

provided in Commentary G “Snow Loads” of the NBC 2005 Structural Commentaries.

Snow loads cannot be reduced based on tributary area.

Specified snow load:

[ ( ) ] where

= Importance factor for snow load (see Table 2 and Table 3, p. 3-21) = ground snow load (kPa)

= associated rain load (kPa)

and are determined based on metrological data: depend on location based on 1-in-50 probability of exceedance per year values are given in Appendix C of NBC 2005 (e.g. See Climate Date for

Design of Buildings, p 3-22).

= basic roof snow factor = 0.8 = higher for large roofs (see NBC 2005 4.1.6.2.(2))

= wind exposure factor = 1.0 for “normal” conditions = 0.75 or 0.5 for buildings in open locations containing only scattered obstructions and where the roof is exposed to the wind on all sides

= roof slope factor accounts for reduced accumulation on steep roofs

= 1.0 for = ( ) ⁄ for = 0 for = 1.0 for roof valleys

= accumulation factor accounts for drifting Wind flow over gable, arched or curved roofs, and domes may create

drifting on the leeward side of the roof.



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Accumulations may occur in roof valleys. Roofs, canopies and balconies situated below adjacent roofs are

susceptible to drifting. Projections or obstructions such as parapets, penthouses and HVAC

equipment may create triangular snow drift.

= 1.0 where no drifts occur > 1.0 where drifting may occur

Guidance is provided in the Commentary “Snow Loads” for common cases.

Other situations should be addressed based on experience, field observations or wind tunnel studies.

The unit weight of snow on roofs, , varies from 1.0 to 4.5 kN/m3 in Canada. An average value of 3.0 kN/m3 is used in most cases when calculating snow drift loads.



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The distribution of snow on roofs adjacent to higher ones is assumed to

vary linearly, and thus varies linearly over the length Xd. (0) = represents an upper limit, where F is the greater of:

= 2, or

[ (

)

]

where = 2w – w2/l w = shorter of upper roof plan dimensions l = longer of upper roof plan dimensions



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3.5.1 IMPORTANCE FACTORS – SNOW AND WIND LOAD

Accounts for the consequences of failure as it relates to the limit state and use and occupancy of the building (see NBC 2005 Table 4.1.2.1., 4.1.6.2., 4.1.7.1.)

Table 2 - Importance Categories for Buildings (NBC Table 4.1.2.1.)

Table 3 – Importance Factors for Snow and Wind Load

Importance Category Snow Load Importance Factor, Is Wind Load Importance Factor, Iw

Ultimate Limit State Service Limit State Ultimate Limit State Service Limit State

Low 0.8 0.9 0.8 0.75

Normal 1.0 0.9 1.0 0.75

High 1.15 0.9 1.15 0.75

Post-disaster 1.25 0.9 1.25 0.75



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3.5.2 CLIMATE DATA FOR DESIGN OF BUILDINGS (USE DATA FOR BC)

Tab

le 4

- D

esig

n D

ata

fo

r S

ele

cte

d L

oca

tio

ns in C

an

ad

a (

NB

C 2

005

Ap

p.

C)



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3.6 LOAD DUE TO WIND

Wind load randomly applied dynamic load

When a structure is in the part of the wind, it deflects or stops the wind, converting the wind kinetic energy into the potential energy of pressure, they by creating wind load.

Wind loads on structures are complex and are affected by many factors.

Typical pattern of air flow resulting pressure are shown below.

Air Flow Lines and Resulting Pressures on Structures

A structure must be designed for the net wind load on the building as a whole.

In addition, walls and cladding of the structure must be designed for the net wind pressure on these surfaces.

Wind loads may be approached as either equivalent static loads or using wind tunnel tests and dynamic analyses.



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3.6.1 SPECIFIED WIND LOADING – EQUIVALENT STATIC LOAD

NBC Article 4.1.7.1.: Commentary I – “Wind Loads and Effects” – NBC 2005 Structural

Commentaries.

External Pressure or Suction:

[ ( )]

where

= specified external pressure or suction acting normal to the surface (kPa)

= Importance factor for wind load (see Table 2 and Table 3, p. 3-21)

= reference velocity pressure (kPa)

is determined based on metrological data: depend on location based on 1-in-50 probability of exceedance per year values are given in Appendix C of NBC 2005 (e.g. See Climate Date for

Design of Buildings, p 3-22).

= exposure factor increases with height = gust effect factor

= external pressure/ suction coefficient



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3.7 EARTHQUAKE LOADS AND EFFECTS

The loading procedure by earthquakes is the result of a complex interaction between ground motion and the response characteristics of the structure.

Earthquake forces are caused by inertia of the structure that tried to resist ground motion. The forces are proportional to the mass of the building; the heavier the building, the larger the seismic force that actions on it.

Seismic Forces on a Building

When the Ground Moves

There are three general approaches for conducting a seismic analysis to determine live loads due to earthquakes:

I. Equivalent static Lateral Force Procedure (ELF) II. Modal Response Spectrum Analysis Method

III. Time History or Response History Analysis



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The NBC 2005 provisions for Earthquake Loads and Effects are included in Article 4.1.8 and Commentary J “Design for Seismic Effects.”

Minimum lateral earthquake force, :

( ) where

= earthquake importance factor of the structure

( ) = design spectral response acceleration, expressed as a ratio to gravitational acceleration, for a fundamental period of the structure

Seismicity of Canada

= ductility-related force modification factor reflecting the capability of a structure to dissipate energy through inelastic behavior

= overstrength-related force modification factor accounting for the dependable portion of reserve strength in a structure designed according to these provisions



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3.8 LOADING DUE TO RESTRAINED DEFORMATIONS

Concrete volume changes due to creep, shrinkage, and thermal changes will create restraint forces if deformations are restrained.

Commentary E “Effects of Deformations…” – NBC 2005 Structural Commentaries.

Control joints and expansion joints should be used to minimize stresses induced by volume changes.

3.8.1 UNIFORM TEMPERATURE CHANGE:

3.8.2 TEMPERATURE GRADIENT:



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3.9 STRUCTURAL ANALYSIS

Key step in design process: used to determine forces that members must resist

Assumed knowledge:

Analysis of stress and strain

Analysis of beams and frames

Support reactions Axial force, shear force and bending moment diagrams Deflection calculations

Behavior of continuous beams and indeterminate frames

Structural analysis in ENGR 327 will be limited to first-order, linear-elastic structural behavior

Tools:

Rules for arrangement of live load for continuous beams and slabs

Approximate frame analysis

Beam tables – design handbook

3.9.1 ARRANGEMENT OF LIVE LOADS – SIMPLIFIED APPROACH

For continuous beams and other indeterminate structures, pattern live loading must be considered to find maximum force and moment effects

Governing live load pattern may be determined using: 1. Influence lines

2. “Brute force”

3. CSA Rules for continuous beams (A23.3 Clause 9.2.3)

4. Approximate frame analysis (A23.3 Clause 9.3)



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3.9.1.1 ARRANGEMENT OF LOADS FOR CONTINUOUS BEAMS AND SLABS

CSA A23.3 Clause 9.2.3: For continuous beams and one-way slabs, the arrangements of live and dead loads may be limited to combinations of

a) Factored dead load of the structure and factored permanent

superimposed dead load on all spans, with factored partition load and factored live load on two adjacent spans;

b) Factored dead load of the structure and factored permanent superimposed dead load on all spans, with factored partition load and factored live load on alternate spans; and

c) Factored dead and factored live load on all spans.

3.9.1.2 APPROXIMATE FRAME ANALYSIS



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4 BASIC STRUCTURAL DESIGN RELATIONSHIP

Consider a beam in normal use or “in service” Loads present “in service”

Weight of beam itself self weight Load due to “use and occupancy” live load

(Specified by applicable Building Code)

Effects of loads: bending moment Strength of the beam: moment capacity or moment resistance

If the resistance of the beam exceeds the load effect, then the beam is safe 4.1 FACTORS OF SAFETY The concept of member resistance (R), and imposed load (L) play a pivotal role in the estimation of factor of safety (FS). A factor of safety depicts resistance provided by a structural member (system) for a given load. Essentially, the FS is computed as:

Basic relationship for safety: resistance > load effect With an increase in load and/ or decrease in resistance, the FS may fall below 1 (FS < 1), which is considered to be a failure state has reached. Thus the design objective is to keep the FS > 1. However, this representation of FS is simplistic, such that it assumes there is no uncertainty in the quantification of L and R values. Often, the L and R values are subject to variability (aleatory) type uncertainty due to natural heterogeneity or randomness of the design process or



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material variability, which cannot be reduced. These sources of uncertainty may be listed as follows:

• actual loads may differ from those assumed,

• actual loads may be distributed in a manner different from that assumed,

• the assumptions and simplifications inherent in any analysis may result in calculated load effects-moments, shear, etc-different from those that act in the structure,

• the actual structural behavior may differ from that assumed, owing to imperfect knowledge,

• reinforcement may not be in its proper position,

• actual material strength may be different from that specified. As well, the definition and acceptable level of consequence of failure compounds the uncertainty in the design process. Thus, each source of uncertainty has to be considered in the quantification of L and R. Let’s assume that the uncertainty in the L and R are provided through a probability distribution.

System load and capacity distribution In this design method, the concept of safety factor is expressed in terms of probability of failure, P(L>R), which is depicted as the shaded area. The objective, then, is to reduce the probability of failure to a sufficiently low value to be acceptable. Safety margin (Y) can be defined as:

f(L

,R)

L, R

System resistance (R)System Load (L)

P(L>R)



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where for Y < 0, failure is said to occur. The safety margin can be shown as:

Safety margin

The function Y has a mean value Ŷ a standard deviation Y. From the above

figure, Ŷ = 0 + Y. The probability of failure Pf is a function of , where is defined as safety index.

The appropriate values of Pf and hence are chosen bearing in mind the consequence of failure.

Consequence of failure Design

Ductile failures with average consequences of failure 3 – 3.5

Sudden failures or failures with serious consequences of failure 3.5 – 4

4.2 PARTIAL SAFETY FACTORS Precise probability density functions of L and R are often not readily available. Thus, the design follows a rational approach given most likely values of L and R, and introducing two sets of partial safety factors. Let’s define the partial resistance and load factors are applied as:

Rn = nominal resistance computed using specified material strengths and dimensions shown on drawings

= resistance factor reflects uncertainties in computing Rn < 1.0

f(Y

)

Safety margin, Y = R - L

Y

Pf = P(R - L) < 0

Ŷ



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In CSA A23.3, it is assumed that the major factors leading to factored

resistance Rf = Rn, are applied to concrete c and steel s.

Li = specified load effects computed using code specified design loads and structural analysis

= load factor reflects uncertainties in computing Li > 1.0 The load effects are defined for live load, dead load, etc with corresponding load

factors of L, D, etc. NBC 2005 requires that the most unfavourable load combination has to be considered in design.

For CSA A23.3, and were based on the assumptions of

1 in 1000 chance of overload 1 in 100 chance of understrength

and were further calibrated depending on failure mode

Ductile failure Pf 1/10,000

Brittle failure Pf 1/100,000

General from:

For bending moments:



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Consider a beam at failure ultimate limit state

Load present at “ultimate” limit state

• Factored self-weight

• Factored live load



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5 LIMIT STATE DESIGN Structures and structural members must be designed as to satisfy the requirements of safety and good serviceability. Safety is ensured if a structure or a structural member can resist, without collapse, all loads and deformations to which it may be subjected during its construction and intended use. Good serviceability requirements are met if a structure or a structural member performs properly during its useful life span. Limit states are defined as states which correspond to the various safety and serviceability conditions for which a structure is designed.

1. Limit states which concern safety are called ultimate Limit States. They correspond to the limit of:

• loss of equilibrium,

• rupture,

• progressive collapse,

• formation of a plastic mechanism,

• instability,

• fatigue.

2. Limit states which concern serviceability are called serviceability Limit States. They correspond to restrictions regarding:

• excessive deflections,

• excessive crack width,

• undesirable vibrations.

3. Fire Resistance

• Structure must have Code specified fire resistance

4. Durability

• Structure must withstand exposure to environment without excessive deterioration



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6 ULTIMATE LIMIT STATE CSA A23.3 Clause 8.1.3

Factored Resistance Factored Load Effect

6.1 FACTORED RESISTANCE

Material resistance factor, , accounts for variations in material strength and cross-section dimensions

Concrete (Clause 8.4.2):

= 0.65

= factored compressive strength

√ = factored tensile strength

Steel Reinforcement (Clause 8.4.3):

= 0.85 for reinforcing bars and embedded steel anchors; = 0.90 for prestressing tendons;

= 0.90 for structural steel.



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6.2 FACTORED LOAD EFFECT

Use Load Factors and Combinations (Clause 8.3.2) CSA A23.3-04 adopts 2005 NBC (included as Annex C)

NBC 2005 Load types and definitions:

D = Dead load (permanent load) L = Live load due to use and occupancy (variable load) S = Load due to snow and rain (variable load) W = Load due to wind (variable load) E = Earthquake load and effects (rare load) H = Load due to lateral earth pressure and groundwater

(permanent load) P = Effects due to prestress (permanent load) T = Effects due to imposed deformations: contraction,

expansion or deflection caused by temperature changes, creep, shrinkage, moisture changes, ground settlement, or combinations thereof.

Principal load – specified variable or rare load that dominates in a given load combination Companion load – specified variable load that accompanies the principal load in a given combination Principal load factor – factor applied to the principal load in a combination to account for the variability of the load, load pattern and analysis of the load effects

Companion load factor – factor applied to the companion load in a combination to give the probable magnitude of a companion load acting simultaneously with the factored principal load



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6.3 FACTORED LOAD COMBINATION FOR ULTIMATE LIMIT STATE

Combination of factored specified principal and companion loads used for the evaluation of an ultimate limit state

Load combination = Factored principal load + Factored companion loads

The factored load effect to be use for design at the ultimate limit state is the combination of factored loads that produces the most critical effect.



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Example A partial floor plan of a reinforced concrete building is shown in the figure below. The roof is subjected to a dead load (D) of 6 kPa (including self weight), a live load (L) of 1.0 kPa and a snow load (S) of 3.0 kPa. Determine the factored axial compression load for a typical interior column C1 supporting the roof according to NBC 2005 requirement.



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7 SERVICEABILITY LIMIT STATE

Conditions in structure checked under service load condition Principal load factors = 1.0

Primary concerns: deflections () and crack widths (w)

< limiting value in code w < limiting value in code

Other concerns: vibrations, permanent deformation LOAD COMBINATIONS FOR SERVICEABILITY

Combination of specified principal and companion loads used for the evaluation of a serviceability limit state

Commentary A “Limit State Design” of NBC 2005 Structural Commentaries

Condition Service load Combination

Long term deflection D + Tp + L + S

Short term deflection L + S

L + S W

Cracking D + Tp + L + S D + Tp + L + W D + Tp + L + Ts

= companion service load factor = 0.5 for Live load = 0.2 for Snow load

8 REFERENCES

1) NRCC, National Building Code of Canada (NBCC 2005), National Research Council Canada, Ottawa, ON, 2005.

2) NRCC, User’s Guide – NBC 2005 Commentaries (Part 4 of Division B), National Research Council Canada, Ottawa, ON, 2005.

3) Brzev, S. and Pao, J. 2006. Reinforced Concrete Design-A Practical Approach, Prentice Hall.

4) MacGregor, J.G. and Bartlett, F.M. 2000. Reinforced Concrete – Mechanics and Design, Prentice Hall, 1st Canadian Edition.

5) Canadian Portland Cement Association 2005. Concrete Design Handbook. Third edition. (Contains the 2004 edition of the design standard for reinforced concrete structures, CSA A23.3-04).

Documents

Chapter 3 - Loads