NFATEC – L17 – Characterisation and idealization of moment...

NFATEC – L17 – Characterisation and idealization of moment resist joint

{LECTURE}

{LTITLE}

Characterisation and Idealisation of Moment Resisting Joints

{/LTITLE}

{LASTEDIT}

/07/04

{/LASTEDIT}

{AUTHOR}

Andy Tyas

{/AUTHOR}

{EMAIL}[email protected]{/EMAIL}

{OVERVIEW}

• The principles of the component method for joint characterisation are presented. This consists of three steps: identification of the components, repsonse of the components and the so-called assembly procedure.

• The idealisation of the moment-rotation characteristic of the joints is introduced: this comprises selecting the shape of the moment-rotation relationship which is the most appropriate to the type of global frame analysis which is planned (elastic, rigid-plastic, …).

• Detailed numerical examples of the use of the component method are beyond the scope of this lecture, but references to relevant sections of EC3 are noted.

{/OVERVIEW}

{PREREQUISITES}

• Basic knowledge on frame analysis and design. • Basic definitions and concepts for joints.

{/PREREQUISITES}

{OBJECTIVES}

The student should:

• Know, at least qualitatively, how to characterise and idealise the response of a structural joint in view of a global frame analysis, following the component method of {ECLINK}Eurocode 3, Part 1.8{/ECLINK}.

{/OBJECTIVES}

{REFERENCES}

• {ECLINK}prEN 1993-1-8: 2003 Eurocode 3 Design of Steel Structures. Part 1-8: Design of Joints. Stage 49 draft (Nov 2003){/ECLINK}.

• Frame design including joint behaviour. User’s manual published by the European Union, Report EUR 18563 EN, Office for Official Publications, Luxembourg, 1998 (ISBN 92-828-4904-X).

{/REFERENCES}

{SECTION}

{STITLE}

General

{/STITLE}

{SUMMARY}

{SUMTITLE}

Characterisation of rotational response of joints

{/SUMTITLE}

An important step when designing a frame consists of the characterisation of the rotational response of the joints, i.e. the evaluation of the mechanical properties in terms of stiffness, strength and ductility.

{DETAIL}

Three main approaches may be followed:

• experimental • numerical • analytical

The only practical option for the designer is the analytical approach. Analytical procedures have been developed which enable a prediction of the joint response based on the knowledge of the mechanical and geometrical properties of the joint components.

In this section a general analytical procedure, termed the component method, is introduced. It applies to any type of steel or composite joints, whatever the geometrical configuration, the type of loading (axial force and/or bending moment, ...) and the type of member sections.

{/DETAIL}

{/SUMMARY}

{/SECTION}

{SECTION}

{STITLE}

Introduction to the component method

{/STITLE}

{TEST}

{TTITLE}

Component method

{/TTITLE}

{QUESTION}

{QTITLE}

Joint characterisation

{/QTITLE}

{QTYPE}MC{/QTYPE}

{QTEXT}

Which of the following statements about joint characterisation and the component method is correct? (More than one may be correct)

{/QTEXT}

{ANSWER}

a) The component method allows designers to break down the complex behaviour of a joint into a series of more readily understandable and calculable sub-parts.

{CHECKMARK}1{/CHECKMARK}

{CHECK}CORRECT - Joint characterisation is the process by which a designer identifies the stiffness, strength and ductility of a joint. The component method allows designers to take a complex joint and break down its behaviour into a set of more easily understandable sub-components, for example column web shear, bolt tension or bearing etc. The behaviour of each of these sub-parts can be identified and quantified, with each having a characteristic strength, and stiffness which may include both elastic and plastic regimes. The performance of the joint as a whole is determined by the interaction of the components.{/CHECK}

{UNCHECKMARK}0{/UNCHECKMARK}

{UNCHECK}INCORRECT - Joint characterisation is the process by which a designer identifies the stiffness, strength and ductility of a joint. The component method allows designers to take a complex joint and break down its behaviour into a set of more easily understandable sub-components, for example column web shear, bolt tension or bearing etc. The behaviour of each of these sub-parts can be identified and quantified, with each having a characteristic strength, and stiffness which may include both elastic and plastic regimes. The performance of the joint as a whole is determined by the interaction of the components.{/UNCHECK}

{/ANSWER}

{ANSWER}

b) The component method can only be applied to joints where the behaviour of each component is assumed to be rigid-plastic.


{CHECK}INCORRECT - The component method can also be applied to cases where linear elastic-plastic or fully non-linear behaviour is assumed{/CHECK}


{UNCHECK}CORRECT - The component method can also be applied to cases where linear elastic-plastic or fully non-linear behaviour is assumed{/UNCHECK}

{/ANSWER}

{ANSWER}

c) To determine the stiffness of the joint as a whole, it is necessary only to find the stiffness of the most flexible sub-component.


{CHECK}INCORRECT - The stiffness of the whole joint depends on the interation of the stiffnesses of the joint componets{/CHECK}


{UNCHECK}CORRECT - The stiffness of the whole joint depends on the interation of the stiffnesses of the joint componets{/UNCHECK}

{/ANSWER}

{ANSWER}

d) The moment resistance of the joint as a whole depends on the strength of the weakest sub-component.


{CHECK}CORRECT - The overall moment resistance of the joint is determined by the strength of the weakest component, whilst the stiffness of the joint is found from a calculation summing the stiffnesses of each individual component.{/CHECK}


{UNCHECK}INCORRECT - The overall moment resistance of the joint is determined by the strength of the weakest component, whilst the stiffness of the joint is found from a calculation summing the stiffnesses of each individual component.{/UNCHECK}

{/ANSWER}

{/QUESTION}

{QUESTION}

{QTITLE}

Joint components

{/QTITLE}

{QTYPE}MC{/QTYPE}

{QTEXT}

Shown below are a list of twelve joint component types covered by {ECLINK}Eurocode 3 Part 1.8{/ECLINK}. Tick those which you think will be active for the joint shown in the figure below if it is to transfer a +ve (hogging) moment between the beam and column.

{IMAGE}

realfigure2g.gif

{/IMAGE}

{/QTEXT}

{ANSWER}1 Column web panel in shear

{IMAGE}realtable1a.gif{/IMAGE}


{CHECK} CORRECT - The +ve bending moment causes tension in the upper part of the joint and compression in the lower part. These forces must be transferred into the column, and the effect of the equal and opposite actions causes shearing of the column web.{/CHECK}


{UNCHECK} INCORRECT - The +ve bending moment causes tension in the upper part of the joint and compression in the lower part. These forces must be transferred into the column, and the effect of the equal and opposite actions causes shearing of the column web.{/UNCHECK}

{/ANSWER}

{ANSWER}2 Column web in compression

{IMAGE}realtable1b.gif{/IMAGE}


{CHECK} CORRECT - The +ve bending moment induced in the column causes compression in the column web adjacent to the lower flange of the beam.{/CHECK}


{UNCHECK} INCORRECT - The +ve bending moment induced in the column causes compression in the column web adjacent to the lower flange of the beam.{/UNCHECK}

{/ANSWER}

{ANSWER}3 Column web in tension

{IMAGE}realtable1e.gif{/IMAGE}


{CHECK} CORRECT - The +ve bending moment induced in the column causes tension in the column web adjacent to the upper flange of the beam.{/CHECK}


{UNCHECK} INCORRECT - The +ve bending moment induced in the column causes tension in the column web adjacent to the upper flange of the beam.{/UNCHECK}

{/ANSWER}

{ANSWER}4 Column flange in bending

{IMAGE}realtable1d.gif{/IMAGE}


{CHECK} CORRECT - The bolts connecting the upper part of the end plate to the column flange will be in tension, to resist the +ve moment. Hence, they will cause the flanges of the column to bend outwards. NOTE: This bending can be reduced or effectively eliminated by adding stiffeners to the column flange. In the example in Figure 2 of the notes, where the beam was welded to the column, the weld between the end of the beam tension flange and the column flange effectively pulls the column flange as a rigid line and thus prevents it bending about the column web.{/CHECK}


{UNCHECK} INCORRECT - The bolts connecting the upper part of the end plate to the column flange will be in tension, to resist the +ve moment. Hence, they will cause the flanges of the column to bend outwards. NOTE: This bending can be reduced or effectively eliminated by adding stiffeners to the column flange. In the example in Figure 2 of the notes, where the beam was welded to the column, the weld between the end of the beam tension flange and the column flange effectively pulls the column flange as a rigid line and thus prevents it bending about the column web.{/UNCHECK}

{/ANSWER}

{ANSWER}5 End-plate in bending

{IMAGE}realtable1f.gif{/IMAGE}


{CHECK} INCORRECT - There is no end-plate in this joint.{/CHECK}


{UNCHECK} CORRECT - There is no end-plate in this joint.{/UNCHECK}

{/ANSWER}

{ANSWER}6 Flange cleat in bending

{IMAGE}realtable1i.gif{/IMAGE}


{CHECK} CORRECT - The tensile force in the upper flange of the beam is transferred to the column flange via bending of the angle cleat.{/CHECK}


{UNCHECK} INCORRECT - The tensile force in the upper flange of the beam is transferred to the column flange via bending of the angle cleat.{/UNCHECK}

{/ANSWER}

{ANSWER}7 Beam or column flange or web in compression

{IMAGE}realtable1c.gif{/IMAGE}


{CHECK} CORRECT - The lower flange of the beam is in compression due to the +ve moment. However, remeber that the value of stiffness for this component may be taken as infinity and thus the component can be safely ignored in detremining the rotational stiffness of the joint.{/CHECK}


{UNCHECK} INCORRECT - The lower flange of the beam is in compression due to the +ve moment. However, remeber that the value of stiffness for this component may be taken as infinity and thus the component can be safely ignored in detremining the rotational stiffness of the joint.{/UNCHECK}

{/ANSWER}

{ANSWER}8 Beam web in tension

{IMAGE}realtable1g.gif{/IMAGE}


{CHECK} INCORRECT - Beam webs are usually put into tension in end-plate connections, where the end-plate is welded to the web and flanges of the beam. In such cases, the bending moment in the beam is transferred into the joint through end-plate, causing tension in both the upper flange and the upper part of the web (assuming a +ve moment). In the case of a flange-cleat connection, all the bending moment in the beam is transferred into the joint through the flanges only.{/CHECK}


{UNCHECK} CORRECT - Beam webs are usually put into tension in end-plate connections, where the end-plate is welded to the web and flanges of the beam. In such cases, the bending moment in the beam is transferred into the joint through end-

plate, causing tension in both the upper flange and the upper part of the web (assuming a +ve moment). In the case of a flange-cleat connection, all the bending moment in the beam is transferred into the joint through the flanges only.{/UNCHECK}

{/ANSWER}

{ANSWER}9 Plate in compression or tension

{IMAGE}realtable1m.gif{/IMAGE}


{CHECK} CORRECT - The horizontal legs of the cleats are considered to be a plate in compression or tension (bottom and top cleats respectively for a +ve moment). However, remeber that the value of stiffness for this component may be taken as infinity and thus the component can be safely ignored in detremining the rotational stiffness of the joint.{/CHECK}


{UNCHECK} INCORRECT - The horizontal legs of the cleats are considered to be a plate in compression or tension (bottom and top cleats respectively for a +ve moment). However, remeber that the value of stiffness for this component may be taken as infinity and thus the component can be safely ignored in detremining the rotational stiffness of the joint.{/UNCHECK}

{/ANSWER}

{ANSWER}10 Bolts in tension

{IMAGE}realtable1j.gif{/IMAGE}


{CHECK} CORRECT - Since the top flange of the beam is in tension due to the hoggon moment, the bolts connecting the angle cleat on that flange to the column flange muxt also be in tension.{/CHECK}


{UNCHECK} INCORRECT - Since the top flange of the beam is in tension due to the hoggon moment, the bolts connecting the angle cleat on that flange to the column flange muxt also be in tension.{/UNCHECK}

{/ANSWER}

{ANSWER}11 Bolts in shear

{IMAGE}realtable1k.gif{/IMAGE}


{CHECK} CORRECT - The forces in both flanges of the beam are transferred into the bolts by bearing of the flanges against the bolts, which themselves then bear against the angle cleats. The bolts themselves are thus put into shear by this equal and opposite bearing action.{/CHECK}


{UNCHECK} INCORRECT - The forces in both flanges of the beam are transferred into the bolts by bearing of the flanges against the bolts, which themselves then bear against the angle cleats. The bolts themselves are thus put into shear by this equal and opposite bearing action.{/UNCHECK}

{/ANSWER}

{ANSWER}12 Bolts in bearing (on beam flange, column flange, end-plate or cleat

{IMAGE}realtable1l.gif{/IMAGE}


{CHECK} CORRECT - The forces in both flanges of the beam are transferred into the bolts by bearing of the flanges against the bolts and then into the cleats again by bearing of the bolts against the holes in the cleats.{/CHECK}


{UNCHECK} INCORRECT - The forces in both flanges of the beam are transferred into the bolts by bearing of the flanges against the bolts and then into the cleats again by bearing of the bolts against the holes in the cleats.{/UNCHECK}

{/ANSWER}

{/QUESTION}

{/TEST}

{SUMMARY}

{SUMTITLE}

Component method - the three step process

{/SUMTITLE}

The application of the component method requires the following steps:

1. Identification of the active components in the joint being considered.

2. Evaluation of the stiffness and/or resistance characteristics for each individual basic component (specific characteristics - initial stiffness, design resistance, or the whole moment-rotation deformability curve).

3. Assembly of all the constituent components and evaluation of the stiffness and/or resistance characteristics of the whole joint.

{DETAIL}

In Figure 1, the principles of the component method are illustrated in the specific case of a beam-to-column joint using H or I sections with a welded connection. For this joint, the relevant components are the column web panel in shear, column web in transverse tension and column web in transverse compression.

The three steps of the component

method

{IMAGE}FinalFIGURE1a.gif{/IMAGE}

Step 1 - Identification

of the components

{IMAGE}FinalFIGURE1b.gif{/IMAGE}

Shear.............Compression..............Tension

Step 2 - Response of

the components

{IMAGE}FinalFIGURE1c.gif{/IMAGE}

Find:

The stiffness coefficients of the joint components {EQN}ki.gif{/EQN}

The resistance of the joint components {EQN}frdi.gif{/EQN}

Step 3 - Assembly of

the components

{IMAGE}FinalFigure1h.gif{/IMAGE}

Overall stiffness of the joint {EQN}sjini1.gif{/EQN}

Overall resistance of the joint {EQN}mrd1.gif{/EQN}

{FIGURE}

Figure 1 Application of the component method to a welded joint

{/FIGURE}

The assembly procedure consists of deriving the mechanical properties of the whole joint from those of all the individual constituent components. This requires a preliminary distribution of the forces acting on the joint into internal forces acting on the components in a way that satisfies equilibrium.

In Eurocode 3 Part 1.8, the analytical assembly procedures are described for the evaluation of the initial stiffness and the design moment resistance of the joint. These two properties enable the designer to determine the design joint moment-rotation characteristic whatever the type of analysis.

{/DETAIL}

{/SUMMARY}

{SUMMARY}

{SUMTITLE}

Joint components

{/SUMTITLE}

The component method considers any joint as a set of individual basic components.

{DETAIL}

For the particular joint shown in Figure 2. (joint with an extended end-plate connection subject to bending), the relevant components are the following:

Compression zone:

• column web in transverse compression • beam flange in compression • beam web in compression

Tension zone:

• column web in transverse tension • column flange in bending • bolts in tension • end-plate in bending • beam web in tension

Shear zone:

• column web panel in shear

{IMAGE}

FinalFigure2.gif

{/IMAGE}

{FIGURE}

Figure 2 Example of a joint covered by Eurocode 3 - Extended end-plate connection

{/FIGURE}

Each of these basic components possesses its own strength and stiffness either in tension or in compression or in shear. The column web is subject to coincident compression, tension and shear. This coexistence of several components within the same joint element can obviously lead to stress interactions that are likely to decrease the resistance of the individual basic components.

The application of the component method requires the designer to identify the joint components. Some of those covered by Eurocode 3 are listed in Table 1. The combination of these components allows one to cover a wide range of joint configurations, which should be sufficient to satisfy the needs of practitioners as far as beam-to-column joints and beam splices in bending are concerned. Examples of such joints are given in Figure 3 at the end of this section.

Other components are also included in Eurocode 3 Part 1.8, such as:

• Haunched beams • Column bases

For simplicity, these latter situations are not covered in this lecture.

{IMAGE}realtable1a.gif{/IMAGE} {IMAGE}realtable1c.gif{/IMAGE}

1. Column web panel in shear 7. Beam or column flange or web in compression

{IMAGE}realtable1b.gif{/IMAGE} {IMAGE}realtable1g.gif{/IMAGE}

2. Column web in transverse compression

8. Beam web in tension

{IMAGE}realtable1e.gif{/IMAGE} {IMAGE}realtable1m.gif{/IMAGE}

3. Column web in transverse tension

9. Plate in tension or compression

{IMAGE}realtable1d.gif{/IMAGE} {IMAGE}realtable1j.gif{/IMAGE}

4. Column flange in bending 10. Bolts in tension

{IMAGE}realtable1f.gif{/IMAGE} {IMAGE}realtable1k.gif{/IMAGE}

5. End-plate in bending 11. Bolts in shear

{IMAGE}realtable1i.gif{/IMAGE} {IMAGE}realtable1l.gif{/IMAGE}

6. Flange cleat in bending 12. Bolts in bearing (on flange,

cleat etc)

{FIGURE}

Table 1 List of components covered by Eurocode 3 Part 1.8

{/FIGURE}

{IMAGE}FinalFigure3a.gif{/IMAGE} {IMAGE}FinalFigure3e.gif{/IMAGE}

(a) - Welded joint (e) - End-plate type beam splice

{IMAGE}FinalFigure3b.gif{/IMAGE} {IMAGE}FinalFigure3f.gif{/IMAGE}

(b) - Bolted joint with extended end-plate

(f) - Cover-plate type beam splice

{IMAGE}FinalFigure3c.gif{/IMAGE} {IMAGE}FinalFigure3g.gif{/IMAGE}

(c) - Double-sided joint with flush end-plates

(g) - Bolted joint with angle flange cleats

{IMAGE}FinalFigure3d.gif{/IMAGE} {IMAGE}FinalFigure3h.gif{/IMAGE}

(d) - Single-sided joint with flush end-plate

(h) - Double-sided beam-to-column joints

{FIGURE}

Figure 3 Examples of joints covered by Eurocode 3

{/FIGURE}

The basic components which are relevant for each type of joint are listed in Tables 6.9 and 6.10 of EC3 Part 1.8. It should be noted that the following components are not specifically required to be taken into consideration in determining the rotational stiffness of the joint, as they can reasonably be assumed to be infinitely stiff.

Component 7. Beam flange and web in compression

Component 8. Beam web in tension

Component 9. Plate in tension or compression

Of course, the analyst must still consider these components (if they are suitable for a given joint type) when determining the moment resistance of the joint.

{/DETAIL}

{/SUMMARY}

{/SECTION}

{SECTION}

{STITLE}

Stiffness and resistance properties of joints

{/STITLE}

{TEST}

{TTITLE}

Stiffness and resistance properties of joints

{/TTITLE}

{QUESTION}

{QTITLE}

Characterisation of the response of a joint in bending

{/QTITLE}

{QTYPE}MC{/QTYPE}

{QTEXT}

Which ONE of the following pairs of factors are, according to EC3, important parameters in characterisation of the response of a joint in bending:

{/QTEXT}

{ANSWER}(a) Maximum joint rotation at failure and initial elastic joint stiffness.


{CHECK}INCORRECT{/CHECK}


{UNCHECK}CORRECT{/UNCHECK}

{/ANSWER}

{ANSWER}(b) Initial elastic joint stiffness and design moment of resistance.


{CHECK}CORRECT{/CHECK}


{UNCHECK}INCORRECT{/UNCHECK}

{/ANSWER}

{ANSWER}(c) Design moment of resistance and secant stiffness of the joint at this value of moment.





{/ANSWER}

{ANSWER}(d) Maximum joint rotation at failure and secant joint stiffness at the design moment of resistance.





{/ANSWER}

{/QUESTION}

{/TEST}

{SUMMARY}

{SUMTITLE}

Stiffness and resistance of joint components

{/SUMTITLE}

The second step of the component method is to determine the stiffness and strength of each component of the joint. The resistances and stiffness coefficients of the individual components may be determined according to Clauses 6.2.6 and 6.3.2 of EC 3 Part 1.8 respectively.

Description of the full method for determining the values of stiffness coefficients and resistances of every component is beyond the scope of this lecture.

{/SUMMARY}

{SUMMARY}

{SUMTITLE}

Internal forces within joints

{/SUMTITLE}

The elastic behaviour of each component is represented by an extensional spring. The force-deformation relationship of this spring is given by:

{EQN}

eq1.gif

{/EQN}

{FIGURE}

(1)

{/FIGURE}

Where {EQN}E.gif{/EQN} is Young's modulus and {EQN}fi.gif{/EQN}is the force, {EQN}ki.gif{/EQN} is the stiffness coefficient and {EQN}Deltai.gif{/EQN}is the deformation of the spring i.

Application of the component method to determine the values of stiffness coefficients and resistance of the component parts of the joint therefore requires the analyst to have an understanding of the distribution of internal forces within the joint.

{DETAIL}

The procedure to find the internal forces within structural joints is quite similar to that used to determine internal stresses in beam and column cross-sections. In the next paragraphs, the procedure followed by Eurocode 3 Part 1.8 is described; it applies to steel beam-to-column joints and beam splices where the beam(s) is (are) subject to bending moments and shear forces. For sake of simplicity and to allow for a hand calculation, two separate distribution procedures are detailed, one for the evaluation of the elastic initial stiffness and the other for the assessment of the design resistance of the joint.

The initial elastic stiffness and the design resistance are considered by Eurocode 3 as the two main parameters characterizing the response of a joint in bending. Based on these two values, a full moment-rotation curve can then be derived as shown in Figure 4.

Provided that the non-linear moment-rotation curve of the joint is not limited by the rotational capacity, this curve consists of three parts. Up to a level of 2/3 of the design moment resistance {EQN}mrd.gif{/EQN}, the curve is assumed to be linear elastic and the corresponding stiffness is the so-called initial stiffness {EQN}sjini.gif{/EQN}. Between 2/3 {EQN}mrd.gif{/EQN} and {EQN}mrd.gif{/EQN}, the curve is non-linear. Once the moment in the joint reaches {EQN}mrd.gif{/EQN}, a yield plateau develops under further imposed rotations of the joint.

{IMAGE}

FinalFigure4.gif

{/IMAGE}

{FIGURE}

Figure 4 Non-linear moment-rotation curve according to Eurocode 3 Part 1.8

{/FIGURE}

The model assumes a fixed ratio between the initial stiffness {EQN}sjini.gif{/EQN} and the secant stiffness at the intersection between the non-linear part and the yield plateau {EQN}sj.gif{/EQN} at level {EQN}mrd.gif{/EQN}. For end-plate and welded joints, this ratio is equal to 3. For flange cleat joints, this ratio is 3,5.

The shape of the non-linear part between 2/3 {EQN}mrd.gif{/EQN} and {EQN}mrd.gif{/EQN} is given by the following interpolation formula:

{EQN}

eq2.gif

{/EQN}

{FIGURE}

(2)

{/FIGURE}

Where {EQN}psi.gif{/EQN} = 2,7 for end-plate and welded joints and 3,1 for flange cleat joints.

{/DETAIL}

{/SUMMARY}

{/SECTION}

{SECTION}

{STITLE}

Stiffness assembly

{/STITLE}

{TEST}

{TTITLE}

Stiffness assembly

{/TTITLE}

{QUESTION}

{QTITLE}

Units of rotational stiffness

{/QTITLE}

{QTYPE}MC{/QTYPE}

{QTEXT}

The rotational stiffness of a joint is expressed in terms of which of the following:

{/QTEXT}

{ANSWER}(a) N/m/rad





{/ANSWER}

{ANSWER}(b) N/m2





{/ANSWER}

{ANSWER}(c) Nm/rad





{/ANSWER}

{ANSWER}(d) Nm/m





{/ANSWER}

{/QUESTION}

{QUESTION}

{QTITLE}

Spring models

{/QTITLE}

{QTYPE}MC{/QTYPE}

{QTEXT}

Figure 1 shows a spring model for an unstiffened welded beam-column joint. Which ONE of the following statements is correct?

{IMAGE}

FinalFigure5.gif

{/IMAGE}

{FIGURE}

Figure 1 Spring model for an unstiffened welded joint

{/FIGURE}

{/QTEXT}

{ANSWER}(a) The force in all the spring components must be equal.





{/ANSWER}

{ANSWER}(b) The deflection of all the spring components must be equal.





{/ANSWER}

{ANSWER}(c) The overall stiffness of the joint is equivalent to the sum of the component stiffnesses.





{/ANSWER}

{ANSWER}(d) The overall rotation of the joint is given by the sum of the deflection the three springs.





{/ANSWER}

{FEEDBACK}

The joint must resist a moment {EQN}M.gif{/EQN} . The resistance is generated, in this type of joint by equal and opposite tension and compression forces {EQN}Ftc.gif{/EQN} in the top and bottom flanges, separated by a lever arm distance {EQN}z.gif{/EQN} . Consider the compression force; this must pass through both springs {EQN}k1.gif{/EQN} and {EQN}k2.gif{/EQN} , and hence the springs must both carry a force {EQN}f.gif{/EQN} . The force in the tension flange must pass through {EQN}k4.gif{/EQN} and hence, this spring too must support a force {EQN}f.gif{/EQN} . Note that the deflections of the three springs are entirely unlinked, and can be derived for each spring from the following equation: {EQN}fi1.gif{/EQN}

{/FEEDBACK}

{/QUESTION}

{QUESTION}

{QTITLE}

Calculating the spring stiffness of a joint

{/QTITLE}

{QTYPE}N{/QTYPE}

{QTEXT}

An unstiffened welded joint connects a beam to a column. The depth between the centres of the beam flanges is 250mm. The stiffness coefficients of the column web panel in shear {EQN}k1.gif{/EQN} , column web in compression {EQN}k2.gif{/EQN} and column web in tension {EQN}k4.gif{/EQN}, are 3mm, 8mm and 8mm respectively and the elastic modulus of steel may be taken as 210kN/mm2. Calculate the overall rotational stiffness of the joint according to the component method of Eurocode 3 (to 3 significant figures).

{/QTEXT}

{ANSWER}

kNm/rad

{VARMIN}22500{/VARMIN}

{VARMAX}22500{/VARMAX}

{/ANSWER}

{FEEDBACK}

Calculation:

{EQN}sjini1.gif{/EQN}

{EQN}Test1.gif{/EQN}

{/FEEDBACK}

{/QUESTION}

{/TEST}

{SUMMARY}

{SUMTITLE}

Assembly of stiffness components

{/SUMTITLE}

The previous sections have considered identification and determination of the values of the joint components. Finally, the spring components in a joint are combined into a spring model representing the overall joint behaviour.

This section will concentrate on examples of how the joint component stiffness coefficients are combined to find an overall stiffness coefficient for the joint. Strength assembly will not be covered here.

{DETAIL}

Figure 5 shows for example the spring model for an unstiffened welded beam-to-column joint.

{IMAGE}

FinalFigure5.gif

{/IMAGE}

{FIGURE}

Figure 5 Spring model for an unstiffened welded joint

{/FIGURE}

Here, the subscripts 1, 2 and 3 refer to the joint components for column web in shear, column web in compression and column web in tension respectively.

The force in each spring i is equal to {EQN}F.gif{/EQN} . The moment {EQN}M.gif{/EQN} acting on the spring model is equal to {EQN}Fz.gif{/EQN} , where {EQN}z.gif{/EQN} is distance between the centre of tension (for welded joints, located in the centre of the upper beam flange) and the centre of compression (for welded joints, located in the centre of the lower beam flange). The rotation {EQN}phi.gif{/EQN} in the joint is equal to {EQN}Delta1-2-3.gif{/EQN} . In other words:

{EQN}

eq3.gif

{/EQN}

{FIGURE}

(3)

{/FIGURE}

Clearly care must be taken in correctly identifying the signs of each {EQN}Deltai.gif{/EQN}

The same formula applies for an end-plate joint with a single bolt-row in tension and for a flange cleat joint. However, the components to be taken into account are different.

Figure 6.a shows the spring model adopted for end-plate joints with two or more bolt-rows in tension. It is assumed that the bolt-row deformations for all rows are proportional to the distance to the point of compression, but that the elastic forces in each row are dependent on the stiffness of the components. Figure 6.b shows how the stiffness coefficients {EQN}kir.gif{/EQN} of components 3, 4, 5 and 10 are added to an effective spring per bolt-row, with an effective stiffness coefficient {EQN}keffr.gif{/EQN} ({EQN}r.gif{/EQN} is the index of the row number). Figure 6.c indicates how these effective springs per bolt-row are replaced by an equivalent spring acting at a lever arm {EQN}z.gif{/EQN} . The stiffness coefficient of this effective spring is {EQN}keq.gif{/EQN} . The effective stiffness coefficient {EQN}keq.gif{/EQN} can directly be applied in Equation 3. The formulae to determine {EQN}keffr.gif{/EQN}, {EQN}z.gif{/EQN}, and {EQN}keq.gif{/EQN} are as follows:

{EQN}

eq4.gif

{/EQN}

{FIGURE}

(4)

{/FIGURE}

{EQN}

eq5.gif

{/EQN}

{FIGURE}

(5)

{/FIGURE}

{EQN}

eq6.gif

{/EQN}

{FIGURE}

(6)

{/FIGURE}

They can be derived from Figure 6. The basis for these formulae is that the moment-rotation behaviour of each of the systems in Figure 6.a to 6.c are equal. An additional condition is that the compressive force in the lower rigid bar is equal in each of these systems.

{IMAGE}

FinalFigure6.gif

{/IMAGE}

{FIGURE}

Figure 6 Spring model for a beam-to-column end-plated joint with more than one bolt-row in tension

{/FIGURE}

In this stiffness model:

• the internal forces are in equilibrium with the bending moment; • the compatibility of the displacements is ensured through the assumption of an

infinitely rigid transverse stiffness of the beam cross-section; • the plasticity criterion is fulfilled as long as the elastic resistance of the springs

is not reached; • no ductility requirement is likely to limit the deformation capacity of the

springs in the elastic range of behaviour as far as Eurocode 3 Part 1.8 components are concerned.

The solution provided by Eurocode 3 Part 1.8 for the initial stiffness prediction fulfils the four main requirements which any distribution of internal forces should satisfy, from a theoretical point of view, and can therefore be considered as an “exact” one.

{/DETAIL}

{/SUMMARY}

{/SECTION}

{SECTION}

{STITLE}

Joint idealisation

{/STITLE}

{TEST}

{TTITLE}

Joint idealisation

{/TTITLE}

{QUESTION}

{QTITLE}

Moment-rotation relationships

{/QTITLE}

{QTYPE}MC{/QTYPE}

{QTEXT}

Figure 1 shows the actual moment-rotation relation for a steel joint, together with a simple idealised elastic-plastic moment-rotation. Which one of the following statements is TRUE of the treatment of this relation by Eurocode 3?

{IMAGE}

realfigure3c.gif

{/IMAGE}

{FIGURE}

Figure 1 Bi-linearisation of a moment-rotation curve

{/FIGURE}

{/QTEXT}

{ANSWER}(a) Characterisation of the joint must exactly describe the non-linear nature of the moment-rotation relation.





{/ANSWER}

{ANSWER}(b) Strain hardening of the joint beyond the maximum elastic moment of resistance must be considered.





{/ANSWER}

{ANSWER}(c) The joint is assumed to have linear stiffness for moments up to its design moment resistance and behave as a perfect plastic hinge thereafter.





{/ANSWER}

{ANSWER}(d) The joint is assumed to be rigid for moments up to its design elastic moment, and behave as a perfect plastic hinge thereafter.





{/ANSWER}

{FEEDBACK}

In figure 1, the dotted line is the actual moment-rotation behaviour, whilst the solid line shows the elastic-plastic idealisation. In the first part of the idealised relation (for moments lower than the design moment resistance, {EQN}mrd.gif{/EQN}, the joint is assumed to rotate with constant stiffness, {EQN}sjini.gif{/EQN}/{EQN}eta.gif{/EQN}, whilst, for higher rotations the joint behaves purely plastically, and further rotation can take place without any increase in moment. This is a simplification of the actual non-linear behaviour, and ignores strain-hardening after the joint has yielded.

{/FEEDBACK}

{/QUESTION}

{QUESTION}

{QTITLE}

Moment-rotation idealisation (Part 1)

{/QTITLE}

{QTYPE}P{/QTYPE}

{QTEXT}

The angle of the straight line idealisation of the moment-rotation relation in Figure 2 makes an angle of {EQN}sjini.gif{/EQN}/{EQN}eta.gif{/EQN} with the rotation axis. Define the two terms in this expression.

{IMAGE}

realfigure4c.gif

{/IMAGE}

{FIGURE}

Figure 2 Linear representation of a moment-rotation curve

{/FIGURE}

{/QTEXT}

{FEEDBACK}

This value defines a hypothetical constant stiffness for the moment-rotation relation, in place of the actual non-linear relation. The value of {EQN}sjini.gif{/EQN}/{EQN}eta.gif{/EQN} is somewhere between the initial stiffness of the joint (i.e., the straight part of the dotted line in Figure 2), and the secant stiffness relative to {EQN}med.gif{/EQN} (the reduced stiffness of the joint when the maximum elastic joint moment resistance {EQN}med.gif{/EQN} is attained). The term {EQN}sjini.gif{/EQN} is the initial stiffness of the joint at low moment and rotation. The term {EQN}eta.gif{/EQN} is a factor by which this value is reduced.

{/FEEDBACK}

{/QUESTION}

{QUESTION}

{QTITLE}

Moment-rotation idealisation (Part 2)

{/QTITLE}

{QTYPE}N{/QTYPE}

{QTEXT}

What value does Eurocode 3 Part 1.8 give for {EQN}eta.gif{/EQN} for typical beam-to column connections?

{/QTEXT}

{ANSWER}

{VARMIN}2{/VARMIN}

{VARMAX}2{/VARMAX}

{/ANSWER}

{/QUESTION}

{/TEST}

{SUMMARY}

{SUMTITLE}

Idealisation of joint moment-rotation characteristics - General principles

{/SUMTITLE}

The non-linear behaviour of the isolated flexural spring which characterises the actual joint response does not lend itself towards everyday design practice. However the moment-rotation characteristic curve may be idealised without significant loss of accuracy.

{DETAIL}

One of the most simple idealisations possible is the elastic-perfectly plastic relationship (shown as the solid line of Figure 7.a). This modelling has the advantage of being quite similar to that used traditionally for the modelling of member cross-sections subject to bending (the solid line of Figure 7.b).

The moment {EQN}mrd.gif{/EQN} that corresponds to the yield plateau is termed the design moment resistance in Eurocode 3. It may be considered as the pseudo-plastic moment resistance of the joint. Strain-hardening effects and possible membrane effects are henceforth neglected, which explains the difference in Figure

7.a between the actual moment-rotation characteristic and the yield plateau of the idealisation.

{IMAGE}

FinalFigure7.gif

{/IMAGE}

{FIGURE}

Figure 7 Bi-linearisation of moment-rotation curves

(a)=Joint, (b)=Member

{/FIGURE}

(NB: In figure 7 and subsequent figures, the dotted, curved lines represent the actual moment-rotation behaviour, whilst the straight lines bounding the red areas represent the idealised behaviour.)

The value of the constant stiffness {EQN}sjini.gif{/EQN}/{EQN}eta.gif{/EQN} is discussed later in this section.

In fact there are different possible ways to idealise a joint moment-rotation characteristic. The choice of one of them is dependent upon the type of frame analysis which is contemplated:

{/DETAIL}

{/SUMMARY}

{SUMMARY}

{SUMTITLE}

1) Elastic idealisation

{/SUMTITLE}

Elastic idealisation of joints is suitable for cases where an elastic global analysis is used.

{DETAIL}

The principal joint characteristic is the constant rotational stiffness.

{IMAGE}

realfigure4b.gif

{/IMAGE}

{FIGURE}

Figure 8 Linear representation of a moment-rotation curve

(a) Elastic verification; (b) Plastic verification

{/FIGURE}

Two possibilities are offered in Eurocode 3 Part 1.8:

• Elastic verification of the joint resistance (Figure 8.a): the constant stiffness is taken equal to the initial stiffness {EQN}sjini.gif{/EQN}; at the end of the frame analysis, a check that the design moment {EQN}med.gif{/EQN} experienced by the joint is less than the maximum elastic joint moment resistance defined as 2/3 {EQN}mrd.gif{/EQN}.

• Plastic verification of the joint resistance (Figure 8.b): the constant stiffness is taken equal to a fictitious stiffness, the value of which is intermediate between the initial stiffness and the secant stiffness relative to {EQN}mrd.gif{/EQN}; it is defined as {EQN}sjini.gif{/EQN}/{EQN}eta.gif{/EQN}. This idealisation is valid for {EQN}med.gif{/EQN} values less than or equal to {EQN}mrd.gif{/EQN}. The values of {EQN}eta.gif{/EQN} are 2 for all beam-to-column connections, 3 for all other welded or bolted end-plate connections and 3,5 for all other bolted cleat connections.

{/DETAIL}

{/SUMMARY}

{SUMMARY}

{SUMTITLE}

2) Rigid-plastic idealisation

{/SUMTITLE}

Rigid-plastic joint idealisation can be used for a rigid-plastic global analysis

{DETAIL}

Only the design resistance {EQN}mrd.gif{/EQN} is needed. In order to allow the possible plastic hinges to form and rotate at the joint locations, it is necessary to check that the joint has a sufficient rotation capacity.

{IMAGE}

FinalFigure9.gif

{/IMAGE}

{FIGURE}

Figure 9 Rigid-plastic representation of a moment-rotation curve

{/FIGURE}

{/DETAIL}

{/SUMMARY}

{SUMMARY}

{SUMTITLE}

3) Non-linear idealisation

{/SUMTITLE}

Non-linear idealisation can be used for an elastic-plastic global analysis

{DETAIL}

The stiffness and resistance properties are of equal importance in this case. The possible idealisations are shown in Figure 10 and range from bi-linear (see Figure 7(a) above) or tri-linear representations to the fully non-linear curve. Again rotation capacity is required in joints where plastic hinges are likely to form and rotate.

{IMAGE}

FinalFigure10a.gif

{/IMAGE}

{FIGURE}

(a) Bi-Linear

{/FIGURE}

{IMAGE}

FinalFigure10b.gif

{/IMAGE}

{FIGURE}

(b) Tri-Linear

{/FIGURE}

{IMAGE}

FinalFigure10c.gif

{/IMAGE}

{FIGURE}

(a) Fully Non-Linear

{/FIGURE}

{FIGURE}

Figure 10 Non-linear representations of a moment-rotation curve

{/FIGURE}

{/DETAIL}

{/SUMMARY}

{/SECTION}

{/LECTURE}

Date post:	14-Apr-2018
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NFATEC – L17 – Characterisation and idealization of moment...

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