STRUCTURAL ANALYSIS
(CIV 4115)
PLASTIC ANALYSIS
UNIVERSITY OF GUYANA
Faculty of Engineering and Technology
Department of Civil & Environmental Engineering
INTRODUCTION
Analysis of structures can be done by the:
• Elastic design method
• Plastic design method
INTRODUCTION
ELASTIC DESIGN METHOD
• Also called the allowable stress method and is based on the elastic
properties of the material.
• Limits the structural usefulness up to the allowable stress.
• Does not account for the material strength pass the elastic stress.
• Heavier design compared to that of the plastic design method.
PLASTIC DESIGN METHOD
• Also called the ultimate load design or load factor design
method and is based on the ultimate load of the material.
• The structural usefulness is maximized by using the strength of
material in the plastic range.
• Smaller design compared to that of the elastic design method.
PURPOSE OF PLASTIC ANALYSIS
• The structure deemed failed at Yield Moment still has capacity to
sustain higher loads and bending moment.
• Extends the Limit state approach to load analysis.
• Actual behaviour of structure beyond the yield point is considered
• Actual behavior of structure at collapse, rather than yield point, is
realistically considered.
PLASTIC MOMENT
• To find the plastic moment of a beam:
Total force in compression = total force in tension
• The ratio of the plastic moment to the yield moment is
known as the shape factor since it depends on the shape of
the cross section.
PLASTIC MOMENT
At a section close to the support where the
bending moment is maximum, the stresses
in the extreme fibers reach the yield stress.
• The moment corresponding to this state
is called the first yield moment My, of
the cross section.
• The moment corresponding to this state
is known as the plastic moment of the
cross section and is denoted by Mp.
PLASTIC MOMENT –
RECTANGULAR SECTIONS
𝑷𝒍𝒂𝒔𝒕𝒊𝒄 𝑴𝒐𝒎𝒆𝒏𝒕, 𝑴𝒑 , 𝒊𝒔:
𝑑
𝑑
𝑀𝑝 = 𝑇 = 𝐶
2
2
=
=
𝐹𝑦 𝑑𝑏
2
𝐹𝑦 𝑑𝑏2
4
×
𝑑
2
SHAPE FACTOR
• The ratio of the plastic moment to the yield moment is
known as the shape factor since it depends on the
shape of the cross section.
Elastic (yield) moment, 𝑀 =
Plastic moment, 𝑀𝑝 =
𝐹𝑦 𝑑𝑏2
4
𝐹𝑦 𝑑𝑏2
6
Shape factor =
𝑀𝑝
𝑀
=
𝐹𝑦 𝑑𝑏2
4
𝐹𝑦 𝑑𝑏2
6
= 1.5
SHAPE
FACTOR
SIGNIFANCE OF SHAPE FACTOR
• It gives an indication of reserve capacity of a section from on set of yielding at
extreme fibres to full plastification.
• If My is known, Mp may be calculated.
• A section with higher shape factor gives a longer warning before collapse.
• A section with higher shape factor is more ductile and gives greater deflection at
collapse.
• The greater is the shape factor value, the greater is collapse load factor
(LF=S x FoS).
LOAD FACTOR
The plastic load factor at rigid plastic collapse (λ) is defined as the
lowest multiple of the design loads which will cause the whole structure,
or any part of it to become a mechanism.
Load factor =
𝑀𝑝
𝑀𝑎
=
𝐹𝑦 𝑍𝑝
𝐹𝑎 𝑍𝑒
𝑀𝑎 − 𝐴𝑙𝑙𝑜𝑤𝑎𝑏𝑙𝑒 𝑚𝑜𝑚𝑒𝑛𝑡
𝑍𝑝 - Plastic section modulus
= 𝑆ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟 × 𝐹𝑜𝑆
𝑍𝑒 - section modulus = 𝑦
𝐼
𝑚𝑎𝑥
PLASTIC HINGE
• A plastic hinge is a zone of yielding due to flexure in a structural member.
• A plastic hinge is similar to a frictionless hinge in that rotation can occur without
any change in bending moment.
• Stresses remain constant at a plastic hinge , but strains and rotations can increase.
• However, it should be recognized that the moment is Mp at plastic hinges rather
than zero at friction less hinges.
PLASTIC HINGE
METHODS OF ANALYSIS
• Static method or Equilibrium method
• Kinematic method or Mechanism method or Virtual work
method
END
CIV 4115 - WIND LOADS EXAMPLE
September 1, 2022
5:51 PM
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STRUCTURAL ANALYSIS
(CIV 4115)
WIND LOADS
UNIVERSITY OF GUYANA
Faculty of Engineering and Technology
Department of Civil & Environmental Engineering
INTRODUCTION
• Analysis of wind loads on structures can be done
using:
• ASCE (American Society of Civil Engineering) 7
Provisions
METHODS
• Wind loads can be determined by using one of the
following methods of analysis:
• Envelope (Simplified) Procedure – for low rise buildings.
• Directional (Analytical) Procedure – for any building height.
• Wind Tunnel Procedure – for all buildings.
EQUATION – WIND PRESSURE
• The basic form of the pressure equation:
p = qGCp – qi(GCpi)
p = wind pressure on a surface
qi = internal velocity pressure.
q = velocity pressure.
Cpi = internal pressure coefficient
G = gust factor
Cp = external pressure coefficient
GUST FACTOR
• Gust Factor (G) accounts for turbulence, gust size and frequency,
structural damping and gust correlation.
• For a rigid building or stiff structures, the gust effect factor is 0.85.
• For a non-rigid (flexible) structure, rotation analysis is applied to
determine this factor.
PRESSURE COEFFICIENT
• Internal Pressure coefficient (Cpi) accounts for
enclosure classification of a building.
• Open
• Ao > 0.8Ag for each wall
• Partially Enclosed
• Ao > 1.10 Aoi, and
• Ao > min[4 sqft , 0.01Ag], and
• Aoi/Agi < 0.20
• Enclosed
• A building that is neither open nor partially enclosed.
Ao = total area of openings in a wall that receives positive external pressure
Ag = the gross area of that wall in which Ao is identified
PRESSURE COEFFICIENT
• External Pressure coefficient (Cp) values can be found below.
VELOCITY PRESSURE
• Velocity pressure (q):
velocity pressure, q = 0.613*Kz*Kzt*Kd*I*V2
Where,
The constant 0.613 is based on the average density of the air at sea level.
VELOCITY PRESSURE
• Velocity pressure (q):
Evaluated at an elevation z:
velocity pressure, qz = 0.613*Kz*Kzt*Kd*I*V2
Evaluated at the building mean roof elevation, h:
velocity pressure, qh = 0.613*Kh*Kzt*Kd*I*V2
VELOCITY PRESSURE DESIGN
PARAMETERS
• V - Basic wind speed (m/s)
• qz - Velocity pressure at height z (N/m2)
• qh - Velocity pressure at mean roof height (N/m2)
• Kd - Wind directionality factor
• Kz – Velocity pressure exposure coefficient
• Kzt – Topographic factor
• I – Importance factor
BASIC WIND SPEED
(V)
• Can be obtained from wind
speed maps.
Wind Speed Maps - ASCE 7-16 (Medeek 2018)
IMPORTANCE
FACTOR (I)
• Category I: I = .87
• Category II: I = 1.00
• Category III & IV: I = 1.15
WIND
DIRECTIONALITY
(Kd)
EXPOSURE CATEGORY
• Exposure is based on ground surface roughness that is determined
from:
• natural topography,
• vegetation, and
• constructed facilities.
EXPOSURE CATEGORY
• Surface Roughness B: Urban and suburban areas, wooded areas, or other terrain with
numerous closely spaced obstructions having the size of single-family dwellings or larger.
• Surface Roughness C: Open terrain with scattered obstructions having heights generally
less than 9.1 m. This category includes flat open country and grasslands.
• Surface Roughness D: Flat, unobstructed areas and water surfaces. This category
includes smooth mud flats, salt flats, and unbroken ice.
TOPOGRAPHIC
FACTOR (Kzt)
• Kzt = 1.0 when:
• H/Lh < 0.2, or
• H < 4.5 m for Exposures C & D, or
• H < 18 m for Exposure B.
• Kzt = (1+K1K2K3)2
TOPOGRAPHIC
FACTOR (Kzt)
• Kzt = 1.0 when:
• H/Lh < 0.2, or
• H < 4.5 m for Exposures C & D, or
• H < 18 m for Exposure B.
• Kzt = (1+K1K2K3)2
VELOCITY EXPOSURE
COEFFICIENT (Kz or Kh)
• 𝐾ℎ is constant and calculated for
mean height of the building
(interpolated from table based on
building height).
• 𝐾𝑧 varies with heights measured
from the base of the building.
PROCEDURE FOR ANALYSIS
1. Define wind speed.
2. Define risk category.
3. Identify importance factor (I) based on risk category.
4. Determine wind load parameters (Kh , Kz , Kd and Kzt).
5. Determine pressure coefficients (Cp and Cpi).
6. Determine gust factor (G).
7. Find velocity pressures (qh and qz).
8. Find wind pressure (P).
PROCEDURE FOR ANALYSIS
• qz acts on the windward wall.
• qh acts on all other walls, leeward and side walls.
EXAMPLE
Calculate the lateral wind loads acting on the 8-story building,
considering the wind is acting first in the North-South direction. The
building is 30 m x 15 m in plan as shown in the figure below and is of a
flat terrain.
Solve using the ASCE Directional Procedure. Assume a 100 km/h basic
wind speed.
END