FRAME STABILITY BY USING EFFECTIVE LENGTH AND SECOND ORDER EFFECT METHOD

NARITA BINTINOH

Msc in Civil Engineering (Structure)

UNIVERSITI TEKNOLOGI MARA(UiTM) MALAYSIA

OCTOBER 2010

ABSTRACT

The structural use of steel in the construction industry is continually growing rapidly across the world as a main construction material. The use of steel as a construction material has its own advantages such as strength, lightness and ductility but it also possesses considerable challenges with regard to slenderness, stability, fire resistance, geometric imperfections and other structural requirements. Thus, single story portal frame offers engineer with high efficiency of structural design. However, current analysis approach still inadequate in examine the stability of the pitched portal frame. Hence there is a need to study on the effect of the pitched angle toward portal frame when it subjected to different types of loading. Therefore, a parametric study on single story of portal frame in sway mode had been carried out to determine the effective length value and second order effect method by using LUSAS modeller (ver. 4.3). The result shows that, effective length by using LUSAS procedure and manual procedure are comparatively the same. However, second order effect method shows that amplification using buckling load approach and national load approach is slightly underestimate compared to second order analysis.

Keywords: steel; second order effect; effective length

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TABLE OF CONTENTS

ABSTRACT

ACKNOWLEDGEMENT

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

CHAPTER 1: INTRODUCTION

1.0 General

1.1 Problem Statement

1.2 Objectives

1.3 Scope of Work And Limitation of Study

1.4 Assumption

CHAPTER 2: LITERATURE REVIEW

2.0 General

2.1 Portal Frame

2.1.1 Arrangement system

2.1.2 Loading system

2.1.3 Connection used

2.2 Method of analysis and geometric non linearity

2.2.1 First order elastic

2.2.2 Second order elastic

2.3 Design approach

2.4 In plane stability check

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2.5 Frame classification

2.6 Buckling load

2.6.1 Computer based

2.6.2 Effective length by using manual procedure

2.6.2.1 Issues in effective length method

2.7 Second order effect

2.7.1 National load approach

2.7.2 Buckling load method

2.7.3 Second order elastic analysis

CHAPTER 3: METHODOLOGY

3.0 Introduction

3.1 Overall Methodology

3.2 Preliminary Studies

3.3 Execution of Modeling in LUSAS

3.3.1 Step of Modeling

3.3.1.1 Description of Geometry Model

3.3.1.2 Meshing

3.3.1.3 Geometric Properties

3.3.1.4 Material Properties

3.3.1.5 Boundary Condition

3.3.2 Loading Input

3.3.3 Analysis Stage

3.4 Data Analysis

3.4.1 Effective length value(manual)

3.4.2 Effective length factor (LUSAS)

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3.4.3 Second order effect (amplification method) 52

3.5 Documentation Process 54

CHAPTER 4: RESULT AND DISCUSSION

4.0 Introduction 55

4.1 Verification Process 57

4.2 Mode of failure 59

4.3 Buckling load 60

4.3.1 Point Load 61

4.3.2 Uniformly Distributed Loading 67

4.4 Effective Length 73

4.4.1 Effective Length Method for Point Load 74

4.4.2 Effective Length Value for Uniform Distribution

Loading 80

4.4.3 Corrected Effective length Value for Uniformly 87

Distributed Loading

4.5 Second Order Effect 93

4.5.1 Point Load 93

4.5.2 Uniformly Distributed Loading 100

CHAPTER 5: CONCLUSSION 108

Recommendation 110

REFERENCES 111

APPENDIX A. 4.1 114

APPENDIX A.4.2 118

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LIST OF TABLES

Descriptions Page

beam column interaction equation for stability check 14 Table 2.1 Table similarities and difference between AISC,CAN, EC3 and AS4100- 18-2.2 1990 19 Table 3.1 Table 3.2 Table 3.3 Table 3.4

Summaries of verification result <-o

Frame dimension that had been developed in L U S A S 40

General stiffness coefficients Kb for beams 49

Approximate values of Kb for beam subjected to axial compression 51

Summary of second order analysis 54

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LIST OF FIGURES

Figure 1.1 Figure 1.2 Figure 1.3 Figure 2.1 Figure 2.2 Figure 2.3 Figure 2.4 Figure 2.5 Figure 2.6

Figure 2.7 Figure 2.8 Figure 2.9 Figure 2.10 Figure 2.11

Figure 2.12 Figure 2.13 Figure 2.14

Figure 2.15

Figure 3.1 Figure 3.2 Figure 3.3 Figure 3.4 Figure 3.5 Figure 3.6 (a) Figure 3.6 (b) Figure 3.7 Figure 3.8 Figure 4.1 Figure 4.2(a-p) Figure 4.3 Figure 4.4

Figure 4.5(a-p)

Figure 4.6

Figure 4.7(a-p)

Descriptions

A Schematic Diagram Of Single Story Portal Frame

Effective length method Second order effect Column bracing in unbraced frames Schematic comparison of load deflection behaviour

Structure behaviour P-delta effect Sway and non sway buckling mode Effective length ratio LE/L for member in rigid jointed frames Alignment chart for determining k-factor in AISC -LFRD Buckled mode for different end connections k- factor for columns with rigid connections Frame deflection; (a) first order; (b) second order Pitched roof frame (a) loading position; (b) anti symmetric buckling mode; (c) symmetric buckling mode Fundamental concept of equivalent initial deflection Portal frame with uniform cross-section Schematic representation of incremental-iterative solution procedure Displacement variable in Updated Lagrarian formulation of a planar beam Frame dimension An overview of the methodology flow chart Flow chart of the verification model process Flow chart of execution modeling process

Modeling in LUSAS Indicating the position of point load Indicating Position of UDL Distribution factors for continuous column Effective length factor, k for columns of sway frames Typical anti symmetric mode of failure Buckling of a frame for each frame configuration Normalization buckling load and effective length Relationship of normalize buckling load with respective of pitched angle Buckling load with respect to pitched angle for uniform distributed load Ratio of buckling load to zero angle frames with respect to pitch angle Normalize effective length value versus pitched angle

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41 42 43 44 45 47

47 49 50

59-60 61-66

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Figure 4.8 Effect of k value with respect to pitch angle for entire 80 model

Figure 4.9(a-p) Normalize k-effective length value versus pitched angle 81 -85 Figure 4.10 Overall effect of k-value normalize toward pitched angle 86 Figure 4.11 Summaries of beam stiffness due to axial forces 87 Figure 4.12(a-p) Comparison of effective length value for LU AS A, 87-92

MANUAL & CORRECTED versus pitched angle Figure 4.13(a-e) Relationship of kamp with respect to Xcrit according to 94-96

pitch angle Figure 4.14(a-e) Relationship of normalized kamp with respect to )xrit 97-99

according to pitched angle Figure 4.15 Relationship between normalized kamp with respect to 100

Xcrit for all frame Figure 4.16(a-e) Relationship between normalize kamp with respect to terit 101-103

accordance to pitched angle Figure 4.17 Relationship between normalize kamp with respect to \crit 107

for all frame

Vlll

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CHAPTER 1

INTRODUCTION

1.0 General

The structural use of steel in the construction industry is incessantly rising rapidly

across the world as a main construction material compared to others. The use of steel as a

construction material has its own advantages and disadvantages. Steel offers designers

with relatively high strength, lightness and ductility. However, it also possesses

substantial challenges with regard to slenderness, stability, fire resistance, geometric

imperfections and other structural requirements (Wang, 2008). Thus, steel framing are

very common to the construction industry players especially in portal frame due to their

high structural efficiency compared to other material.

Portal frame can be define as a structural framing system consisting of a girder

supported by two stanchions to which it is connected with sufficient rigidity to hold

unchanged the original angles between the intersecting members as drawn in Figure 1.1.

The connections between the columns and the rafters are designed to be rigid connection

where they can carry bending moment and evade the building leaning over or racking in

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the width due to sideways forces. Then it leads to the reduction of size of the rafters and

the span can be increased for the same size rafters. This makes portal frames a very

efficient construction technique to use for wide span building in warehouses, barns and

pavilion. Portal frame can be constructing as single story or multi story depending on the

intended design and it was likely design to have pitched angle in order to allow the

movement of water on the roof.

Figure 1.1: A Schematic Diagram of Single Story Portal Frame

Furthermore, portal frame are commonly design and constructed by having

slender column and with wider bay width. Then, this particular structural frame needs to

be check for the structural stability that involved of interaction between beam column

behaviors as a whole system. Therefore, column buckling resistance need to be properly

checked by calculating effective length method or second order analysis. Since, a past

decade there are restriction and limitation on the technology wise, therefore effective

length have been used widespread throughout the world. Since, portal frame is a

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combination of beam and column structures element; hence the effective length can be

determined by considering the effect of the beam column interaction effect by equating

the structural stiffness. For the simplicity, code of practice such as EC3, AISC and

BS5950 provide charts and monograph to find the effective length value according to

sway and non sway condition. On the other hand, the chart was developing for

rectangular frame but not for pitched portal frame. Thus, chart provided by the code of

practice not be able to solve the entire problem regarding the frame stability for pitched

portal frame due to the difficulty of finding the effective length, k as in Figure 1.2.

—̂ -

k L

Figure 1.2: Effective length method

In addition, it is impossible to have an ideal cross section without any

imperfection in straightness in structural members. Hence approximate second order

analysis has been produce by previous researchers in order to encounter the problem

regarding the imperfection and well knows as P-A effects that lead to the additional

secondary overturning moment as in Figure 1.3, whereby the first order moment was not

valid anymore. Since, the technology becomes more advanced and quite a number of

software has been developed in order to simplify the calculation of second order analysis

by providing amplification value, kamp. However most of the research was done on

multistory frame and the roof is orthogonal but very rare on pitched portal framed. Thus,

it is suggested to develop appropriate analysis procedures to quantify the relevant effects

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in the structures and clarify several issues concerning the geometrically nonlinear global

behavior of plane pitched-roof frames by using different types of loading.

->>^

H A

M\storder = H L

M2ndorder=HL + PA

Figure 1.3: Second order effect

1.1 Problem Statement

Single story portal frame offers engineer with high efficiency of structural design

in terms of bay width, member sizes and column height. Even though the effective length

method is available in the different codes of practice but it was based on the assumption

as mention above. Therefore, chart and monograph are not adequate to solve the entire

engineering problem regarding the column buckling resistance in frame system. Hence

second order effect method was introduced to incorporate with the P-A effect. However,

most of the study conducted in previous investigation by other researchers only mean for

orthogonal portal frame not pitched roof frame. Hence there is a need to study on the

effect of the pitched angle toward portal frame when it subjected to different types of

loading. Therefore, a parametric study on single story of portal frame in sway mode had

been carried out to determine the effective length value and second order effect method

by using LUSAS modeller (ver. 4.3).

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L2 Objective

The objectives of this study are:

i. To d e t e n t e the effect of pitch angle on the elastic buckling load of

portal frame by using LUSAS

"• To determine the effect of pitched angle on second order effect of

portal frame by using LUSAS

i". To compare effective length and second order effect between LUSAS

and hand calculation using BS5950.

1-3 Scope of Work and Limitation of Study

This study was conducted on frame stability by adopting various methods which

are second order effect method and effective length method of portal fmme with

pa-uetric study on coluum height, bay span and pitched angle. The entire section

member was determined by adopting reeommeuded size from PERWAJA catalogue.

Hence hand calculation and modeling stimulation by using LUSAS had been carried out

and the finding will be discussed m the Chapter 4. However, this study is valid for the

elastic portion of analysis with uniform structural cross section using umversal beam

provided with the mode of buckling is only for i„ p l a n e b u c k l i n g i n v o | v e d rf ^ ^

However, the comparison only made for pitched augle but not for columu heigh, and bay

width due to different member sizes adopted in each frame.

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1.4 Assumption

This study had been conducted by assuming:

i. The behaviour of the frame is elastic

ii. The frame used is moment resisting frame with rigid joint,

iii. Frame experiencing in plane sway mode.

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CHAPTER 2

LITERATURE REVIEW

2.0 General

By definition s t r u c t u r a l f r a m e s w c o m p o s e d rf o n e . d i m e n s i o n a | ^ ^

connected together in skeletai arrangements which transfer the applied loads to the

snppotts (Trahair et. a!; 2008). Furthermore, frame can be divided into three which are

triangulate frame, two dimensional frame and three dimensional frames. Those frames

are classified based on structural behavior. However, mnlti storey or mnlti bays of

orthogonal frame or pitched fames was fall ,„ the two dimensional frames which are

substantial to bending action and axial forces in structural member. Therefore, father

discussion only covered on behavior and stability aspect of two dimensional frames

primary on portal frame that subjected to the in plane loading only.

2.1 Portal frame

Portal fame broadly used for industrial, commercial and leisure buildings such as

pavilion and stadium which covers large areas without intermediate columns. According

.0 Kumar and Santha (2010) portal frame commonly constructed by using hot-rolled

sections, supporting the roofing and side cladding via eold-formed purlins and sheeting

mils. In addition, Saka (2003) claimed that portal frame build by using steel as a

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construction material involved lower construction cost compared to other material due to

the its own material properties.

In extension, there are a lot of design concept was being adopted for construction

of a portal frame in order to reduce the sizes of the structural member and cut down the

construction cost. For an instance, most of the portal frame design was involved of

tampered structure rather than uniform cross section as mention in PERWAJA catalogue.

Since, the loading from the top are considerably bigger, therefore usage of plate element

also adopted for rafter fabrication. However, since this research was limited to uniform

cross section member, therefore the effect of tempered section and plate element will not

discuss further in this chapter toward portal frame behavior.

Meanwhile, Saka (2003) suggested having four different form of pitched portal

frame design as pitched roof portal frame with pinned-base stanchions, pitched roof

lattice girder with fixed-base stanchions, pitched roof trusses with fixed-base stanchions

and space trusses with fixed-base stanchions. However, pitched roof portal frame with

pinned-base stanchions was frequently adopted in construction site due to economical

reason compared to fixed support system.

Knowing that, the behaviour of portal frame is dependent to it arrangement

system, loading system, and types of connection used in the whole framing system due to

loading transferred from structural member to the support system. Hence all of the three

factors will be discuss further in next sub topic.

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2.1.1 Arrangement system

Arrangement system includes either it was design as single bay,

multi storey or both. This is because, the arrangement of the frame system

may manipulate the buckling strength of the frame system as mention by

Yura (1971), which multi bay frame experiencing higher buckling strength

compared to single bay frame condition as drawn in Figure 2.1.

Figure 2.1: Column bracing in unbraced frames

(Source: Yura; 1971)

Yura (1971) explained that all columns in a frame system must

sway concurrently as recommended in the BS5950, EC3 and AISC. Based

on Figure 2.1, it can be said that if the two exterior columns reach the

buckling reached their individual column buckling load; the whole system

will not buckle. This is because the exterior column did not experience

independent buckling load yet, thus it will contribute to the shear

resistance on exterior column. Hence, lead to the stabilizing effect, where

the exterior columns act as brace system for the structure and effective

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length of some of the columns could be reduced to 1.0. Furthermore, it is

recommended to treat each column individually by assuming the frame are

rigidly jointed and alignment chart can be adopted for calculating the total

buckling strength of a system. Thus, shows that sidesway buckling can be

considered as a total story phenomenon and not as individual failure.

In general, sidesway will not occur until the total frame load on a

story achieves the sum of the potential individual column loads for the

unbraced frame. Thus, shows that single story with single bay of portal

frame can be considered as worse case in determining the effective length

value because they only stand with the exterior column without any

additional interior column to brace the exterior column.

Loading system

When a frame is or can be considered as two-dimensional then its

behaviour is similar to that of the beam-columns of which it is composed.

Therefore the failure was dependent on the position of the loading system

applied either in plane or out of the plane. With in-plane loading only, it

can be predicted to fail either by in-plane bending, or by flexural -

torsional buckling out of its plane. However, this research only concerned

on the in plane behaviour of the frame system when it subjected to load.

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Connection used

There are three common connection used in framing system

especially in portal frame design such as simple connection, semi-rigid

and rigid connection. According to Ashraf et. al. (2007) frames with

simple connections, it was assume to be pinned and the moments

transmitted by the connections are small, and often can be neglected, and

the members can be treated as isolated beams. However, it becomes

contradict when the connections used are semi-rigid or rigid because

moment interactions between the members become significant. Note that,

this research only focused on unbraced single story rigidly connected

portal frame; therefore most of the discussion was based on the rigid frame

behavior.

A rigid joint may be defined as a joint which has sufficient rigidity

to prevent relative rotation between the members connected. Commonly,

proper welded and bolting system is assumed to be rigid. Generally rigid

joint was assumed stiffer and stronger than frames with simple or semi­

rigid joints. Because of this, rigid frames offer significant economies,

while many difficulties associated with their analysis have been greatly

reduced by the availability of standard computer programs (Trahair et al.,;

2008). It is because there a lot of analysis types can be performed in order

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to evaluate the stability of a rigid frame either in elastic region or inelastic

part as discusses in next sub topic.

2.2 Method of analysis and geometric non linearity

There are different levels of analysis with various level or enhancement which can be

used for structural design including linear analysis and second order elastic analysis in

order to incorporate the nonlinearity effect of a steel structure which is geometric non

linearity, material non linearity and non linear force boundary condition as in Figure 2.3.

Since this researched is governed for elastic region, therefore advanced analysis will not

be discussed further in Figure2.2.

2.2.1 First-Order Elastic:

First order elastic analysis considered as the most basic of analysis

method and common approach in routine design, where the

material and geometric are modeled to be linear elastic and

equilibrium is expressed on the undeformed configuration of the

structure. However, first order elastic analysis does not give any

direct measure of frame stability (Galambos; 1997).

2.2.2 Second-Order Elastic

Meanwhile second order analysis consider the material are linear

and geometry are considered to be non linear. Where the

equilibrium is calculated based on deformed geometry of the

structure. In addition load-deflection curve commonly being

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plotted from rigorous iterative method until it reach to a certain

limit, called as elastic stability limit structure. The elastic stability

limit calculated by using second order incremental analysis is

similar but distinct from the elastic critical load (Xcrit) calculated

by a classical stability (eigenvalue) analysis. Differences in two

limits arise because the elastic stability load corresponds to

equilibrium in the deformed configuration, whereas the elastic

critical load is calculated as a bifurcation from equilibrium from

the undefomed geometry. Additionally, elastic loads are usually

calculated for idealized conditions that not represent the actual

loading (Galambos; 1997).

ijl

Lateral Deflection, A

Figure 2.2: Schematic comparison of

load-deflection behavior

(Sources: Galambos; 1997)

ftUBdliBg

. . ;';.'1T - V "'

, ,'" ^(iwmelnciNG-lififflty

[i) Notorial nco-Iirarity

IX'fonnitiori

Figure 2.3: Structure behaviour

(Source: Trahair et. al; 2008)

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2.3 Design approach

In framing system, stability check are crucial in order to ensure the

buckling resistance is more than buckling load either about minor axis or major

axis. Al though present of differences of design approach between different code

of practice but all of them are employ both method either using effective length

method or second order method. If the effective length method are employ

therefore, moment should not be amplified and vice versa in order to fulfill frame

stability checking as in Table 2 . 1 . Thus shows that, code of practice provide two

options in determining the instability check due to limitation of the both method

that will be discussed further.

Table 2 . 1 : beam column interaction equat ion for stability check

Code of practice

BS5950

A I S C - L F R D

EC3

Design approach

Fr mYMr mvM. — 4

lxl Mx + y"y

Pc PyZx PyZ. < 1

4-̂ =1, <Pc Pn(KL) 9 <pbMn (pcPniKL)

Pu +

M» = 1,-

Pu

2(PcPn(KL) <PbMn <PCPn{KL)

>0.2

<0 .2

P' . . . M'x Ml <P'Py

- 0 . 9 0 — ^ - = 1 , — — = 1 <p'Mpx (p'Mpx

M' P' \( P' P'

<p'M py = 1.56 1 — -T5-+ 0 6 ' - T 5 - > 0 2

V <p'PYJ\<pPy J <PPy

My P' —= 1, <0 .2

(p'Mpy (p'Py

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Where

Pu =required axial compressive

P nCKL)= nominal strength of member as concentrically loaded column

Mu = maximum second order elastic moment within span or at ends of

member

Mn = nominal bending capacity of member in absence of axial loading

<Pb> <Pc= resistance factors for axial compression and bending equal to 0.85

and 0.9, respectively.

My, Mx, P = axial force and maximum second order elastic moments

within member about x and y axes

(p'= effective resistance factor

mx, my= equivalent uniform moment factors

Mx=maximum major axis moment in the segment length Lx governing Pcx

My= maximum minor axis moment in the segment length Ly governing Pcy

Fc= axial compression

Z= section modulus

Pcx= compression resistance considering about major axis only

Pcy compression resistance considering about minor axis only

2.4 In plane stability check

All members resisting axial compression would buckle if the applied axial force

were large enough and buckling can be define as failure mode characterized by a

sudden failure of a structural member subjected to high compressive stress,

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where the actual compressive stress at the point of failure is less than the ultimate

compressive stresses that the material is capable of withstanding. This mode of

failure is also described as failure due to elastic instability. Thus, stability check

should be performed in order to ensure the resistance of buckling load is more

than the applied load (Moy; 1991). The checking should include buckling about

major axis and minor axis. However, buckling in the plane of the frame is more

complicated than in normal beam column elements because there is normally no

bracing in the plane of the frame. Kumar and Kumar (undated) together with

Nishino and Ai (2002) explain that, for compression members in rigid-jointed

frames the effective length is directly related to the restraint provided by all the

surrounding members. In a frame the interaction of all the members occurs

because of the frame buckling as a whole rather than single column buckling.

Therefore, checks for the stability of the frame must consider the entire frame

stiffness. The in-plane stability checks for portal frames in BS 5950-1: 2000

differ from those for beam and column buildings due to the axial loads in portal

rafters have a much greater effect on the stability of the frame compared to the

axial loads in common beam and column buildings (King;2001, Silvestre and

Camotim;2006). Although the subject of frame stability is a mature one, the

approaches for consideration of stability in the design of steel frames vary

depending on the specification adopted throughout the world.

Classically, frame stability check was based on effective length value, k

which incorporated with beam and column stiffness. Availability of chart and

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