NFATEC – L11 – Restrained beams (25/02/2004)library.tee.gr/digital/m2247/m2247_l11_eng.pdf · Beam Type Span Range (m) Notes 0. ... Castellated beam{/FIGURE} {IMAGE}Plate girder

$Page 1: NFATEC – L11 – Restrained beams (25/02/2004)library.tee.gr/digital/m2247/m2247_l11_eng.pdf · Beam Type Span Range (m) Notes 0. ... Castellated beam{/FIGURE} {IMAGE}Plate girder$
NFATEC – L11 – Restrained beams (25/02/2004)

{LASTEDIT}Roger 25/02/04{/LASTEDIT}

{LECTURE}

{LTITLE}Restrained Beams{/LTITLE}

{AUTHOR}Roger{/AUTHOR}

{EMAIL}[email protected]{/EMAIL}

{PREREQUISITES}

• Elastic theory for uniaxial bending • Engineering properties of steels • Cross-section classification.

{/PREREQUISITES}

{OBJECTIVES}

On successful completion of this lecture you should:

• be familiar with the procedures used to design restrained beams, • be able to design a beam for bending resistance, • be able to check a beam for compliance with serviceability criteria, • know how to reduce the bending resistance of a beam to allow for high shear

loads.

{/OBJECTIVES}

{REFERENCES}

• prEN 1993-1-1:2003 Eurocode 3: Design of steel structures Part 1-1 General rules and rules for buildings. Stage 49 draft (Nov 2003).

• Trahair, N.S. and Bradord, M.A., The Behaviour and Design of Steel Structures, E & F Spon, 1994.

• Galambos, T.V., Structural Members and Frames, Prentice-Hall, 1968 • Narayanan, R., Beams and Beam Columns - Stability and Strength, Applied

Science, London, 1983

{/REFERENCES}

{OVERVIEW}

• A variety of section shapes are available for beams; choice depends on load and span.

• Beams are generally designed on the basis of bending moment resistance. • Stiffness and its influence on deflection under serviceability loads is an important

consideration. • Beams which are prevented from lateral movement are termed restrained. • Moment resistance is dependent on section classification. • The effect of co-existent shear forces on the moment resistance can be ignored if

they are less than 50% of the plastic shear resistance.

{/OVERVIEW}

{SECTION}

{STITLE}Introduction{/STITLE}

{SUMMARY}

{SUMTITLE}Beam types{/SUMTITLE}

A wide range of different beam types is used for differing applications

{DETAIL}

Beams are perhaps the most basic of structural components. A variety of section shapes and beams types may be used depending on the magnitude of loading and the span, as shown in the table and figures below

Beam Type Span Range

(m)

Notes

0. Angles 3 - 6 used for roof purlins, sheeting rails, etc., where only light loads have to be carried.

1. Cold-formed sections

4 - 8 used for roof purlins, sheeting rails, etc., where only light loads have to be carried.

2. Rolled Sections UB, IPE, UPN, HE

1 - 30 most frequently used type of section; proportions selected to eliminate several possible types of failure.

3. Open web joists 4 - 40 prefabricated using angles or tubes as chords and round bars for web diagonals; used in place of rolled sections.

4. Castellated beams 6 - 60 used for long spans and/or light loads, depth of UB increased by 50%, web openings may be used for services, etc.

5. Compound sections e.g. IPE + UPN

5 - 15 used when a single rolled section would not provide sufficient capacity; often arranged to provide enhanced horizontal bending strength as well.

6. Plate girders 10 - 100 made by welding together 3 plates, sometimes automatically; web depth up to 3-4m sometimes need stiffening.

7. Box girders 15 - 200 fabricated from plate, usually stiffened; used for OHT cranes and bridges due to good torsional and transverse stiffness properties.

{FIGURE}Table 1. Examples of different beam types{/FIGURE}

{IMAGE}Cold formed sections.jpg{/IMAGE}

{FIGURE}Figure 1. Cold formed sections used as purlins{/FIGURE}

{IMAGE}Simple beam.jpg{/IMAGE}

{FIGURE}Figure 2. Simple beam{/FIGURE}

{IMAGE}Castellated beam.jpg{/IMAGE}

{FIGURE}Figure 3. Castellated beam{/FIGURE}

{IMAGE}Plate girder.jpg{/IMAGE}

{FIGURE}Figure 4. Plate girder beam{/FIGURE}

{IMAGE}Box girder.jpg{/IMAGE}

{FIGURE}Figure 5. Box girder beam{/FIGURE}

{/DETAIL}

{/SUMMARY}

{SUMMARY}

{SUMTITLE}Beam restraint{/SUMTITLE}

Restrained beams can be designed simply for bending moment resistance and stiffness. Lateral restraint depends on other elements of construction (continuous or discrete), such as floor slabs or secondary beams, preventing lateral movement or twisting of the cross-section.

{PPT}Lecture11Intro.pps{/PPT}

{DETAIL}

Steel beams can often be designed simply on the basis of bending moment resistance (ensuring the design moment resistance of the selected cross-section exceeds the maximum applied moment) and stiffness, that is the beam does not deflect so much that it affects serviceability considerations. Beams which are unable to move laterally are termed "restrained", and are unaffected by out-of-plane buckling (lateral-torsional instability). Beams may be considered restrained if full lateral restraint is provided by for example positive attachment of a floor system to the top flange of a simply supported beam (many designers consider the friction generated between a concrete slab and steel beam to constitute a positive attachment). adequate torsional restraint of the compression flange is provided, for example by profiled roof sheeting closely spaced bracing elements are provided such that the minor axis slenderness is low (see unrestrained beams for details). Additionally, sections bent about their minor axis cannot fail by lateral torsional instability and it is unlikely that sections with high torsional and lateral stiffness (eg rectangular hollow sections) will fail in this way. The material presented in this lecture assumes adequate restraint of the beams. In practice it is the designer's responsibility to ensure the structural details are consistent with such an assumption

{/DETAIL}

{/SUMMARY}

{/SECTION}

{SECTION}

{STITLE}Moment resistance{/STITLE}

{SUMMARY}

{SUMTITLE}Principles{/SUMTITLE}

The design moment resistance of a cross-section depends the section shape, material strength and section classification.

{PPT}Lecture11MR.pps{/PPT}

{DETAIL}

In a simple single span beam, as shown in the figure below, failure occurs when the design value of the bending moment {EQN}MEd.gif{/EQN} exceeds the design moment resistance of the cross-section, the magnitude of which depends on the section shape, material strength and section classification.

{IMAGE}L11I1.jpg{/IMAGE}

{FIGURE}Figure 6. Behaviour of a simply supported beam showing applied load {EQN}F.gif{/EQN} against vertical deflection {EQN}delta.gif{/EQN}

(a) Elastic (b) Elastic-plastic (c) Plastic{/FIGURE}

For cases where the shear force on the cross-section can be considered small enough that its effect on the resistance moment may be neglected (EC3 sets a shear force value of 50% of the plastic shear resistance) then the design moment resistance {EQN}McRd.gif{/EQN} may be determined as follows:

For class 1 and 2 cross-sections, the design plastic resistance moment of the gross section

{EQN}L11E1.gif{/EQN}

{ECLINK}EC3:Part 1-1:6.2.5(2) (6.13){/ECLINK}

For a class 3 cross-section, the design elastic resistance moment of the gross section



For a class 4 cross-section, the design local buckling resistance



{/DETAIL}

{/SUMMARY}

{TEST}

{TTITLE}Factors influencing moment resistance{/TTITLE}

{QUESTION}

{QTITLE}Moment resistance{/QTITLE}

{QTYPE}MC{/QTYPE}

{QTEXT}

Which of the following parameters affects the bending moment resistance of a cross section?

{/QTEXT}

{ANSWER}

section shape

{CHECKMARK}1{/CHECKMARK}

{CHECK}Yes – section shape is a major influence{/CHECK}

{UNCHECKMARK}0{/UNCHECKMARK}

{UNCHECK}Section shape is a major influence{/UNCHECK}

{/ANSWER}

{ANSWER}

material strength


{CHECK}Yes – material strength is a major influence{/CHECK}


{UNCHECK}Material strength is a major influence{/UNCHECK}

{/ANSWER}

{ANSWER}

section classification


{CHECK}Yes – section classification is a major influence{/CHECK}


{UNCHECK}Section classification is a major influence{/UNCHECK}

{/ANSWER}

{ANSWER}

beam length


{CHECK}No – beam length does not influence the bending moment resistance{/CHECK}


{UNCHECK} Beam length does not influence the bending moment resistance {/UNCHECK}

{/ANSWER}

{ANSWER}

loading


{CHECK}No – loading does not influence the bending moment resistance{/CHECK}


{UNCHECK}Loading does not influence the bending moment resistance {/UNCHECK}

{/ANSWER}

{ANSWER}

support conditions


{CHECK}No – support conditions do not influence the bending moment resistance{/CHECK}


{UNCHECK}Support conditions do not influence the bending moment resistance {/UNCHECK}

{/ANSWER}

{FEEDBACK}The moment resistance of a cross-section is dominated by section shape, material strength and section classification{/FEEDBACK}

{/QUESTION}

{QUESTION}

{QTITLE}Moment-rotation and section classification{/QTITLE}

{QTYPE}M{/QTYPE}

{QTEXT}Match the moment-rotation behaviour illustrated with the section classification{/QTEXT}

{ANSWER}

Section class 1

{MARK}1{/MARK}

{MATCH}


{/MATCH}

{REASON}Class 1 sections are able to develop a plastic hinge with the required rotational capacity for plastic analysis.{/REASON}

{/ANSWER}

{ANSWER}

Section class 2

{MARK}1{/MARK}

{MATCH}


{/MATCH}

{REASON} Class 2 sections are able to develop a plastic hinge but have limited rotational capacity (and are therefore unsuitable for plastic analysis).{/REASON}

{/ANSWER}

{ANSWER}

Section class 3

{MARK}1{/MARK}

{MATCH}


{/MATCH}

{REASON}Class 3 sections are those which are able to develop yield stress at the extreme fibres, but local buckling prevents the development of a plastic hinge.{/REASON}

{/ANSWER}

{ANSWER}

Section class 4

{MARK}1{/MARK}

{MATCH}


{/MATCH}

{REASON}Class 4 sections are those in which local buckling limits the moment resistance and an explicit reduction is necessary.{/REASON}

{/ANSWER}

{/QUESTION}

{QUESTION}

{QTITLE}Section class - plastic resistance and full rotation{/QTITLE}

{QTYPE}N{/QTYPE}

{QTEXT}What class of section can develop plastic bending moment resistance and has full rotation capacity{/QTEXT}

{ANSWER}

{VARMIN}1{/VARMIN}

{VARMAX}1{/VARMAX}

{/ANSWER}

{FEEDBACK} Only class 1 sections are able to develop a plastic hinge with the required rotational capacity for plastic analysis.{/FEEDBACK}

{/QUESTION}

{QUESTION}

{QTITLE}Section class - plastic resistance but limited rotation{/QTITLE}

{QTYPE}N{/QTYPE}

{QTEXT}What class of section may develop its plastic bending moment resistance and has low rotation capacity?{/QTEXT}

{ANSWER}

{VARMIN}2{/VARMIN}

{VARMAX}2{/VARMAX}

{/ANSWER}

{FEEDBACK}Class 2 sections are able to develop a plastic hinge but have limited rotational capacity (and are therefore unsuitable for plastic analysis).{/FEEDBACK}

{/QUESTION}

{QUESTION}

{QTITLE}Section class - elastic resistance and no rotation{/QTITLE}

{QTYPE}N{/QTYPE}

{QTEXT}What class of section can develop its full elastic bending moment resistance and has no rotation capacity?{/QTEXT}

{ANSWER}

{VARMIN}3{/VARMIN}

{VARMAX}3{/VARMAX}

{/ANSWER}

{FEEDBACK}Class 3 sections are able to develop yield stress at the extreme fibres, but local buckling prevents the development of a plastic hinge.{/FEEDBACK}

{/QUESTION}

{QUESTION}

{QTITLE}Section class - limited resistance and no rotation{/QTITLE}

{QTYPE}N{/QTYPE}

{QTEXT}What class of section can develop only limited elastic bending moment resistance (based on an effective cross-section) and has no rotation capacity?{/QTEXT}

{ANSWER}

{VARMIN}4{/VARMIN}

{VARMAX}4{/VARMAX}

{/ANSWER}

{FEEDBACK}Class 4 sections have limited moment resistance based on an explicit reduction for local buckling.{/FEEDBACK}

{/QUESTION}

{/TEST}

{SUMMARY}

{SUMTITLE}Effect of holes{/SUMTITLE}

Design moment capacity can be assumed to be unaffected by bolt holes provided the reduction in cross-section is small.

{PPT}Lecture11Holes.pps{/PPT}

{DETAIL}

If bolt holes are located in the tension flange at the critical cross-section it is required to check that the ratio of the net area to gross area of the flange is not so small that the section would rupture on the net section before the cross section yielded. This check is the same as that given for ductile tension members and will be satisfied provided that

{EQN}L11Eeqn8.gif{/EQN}

{ECLINK}6.2.5(4) (6.16){/ECLINK}

Values of 1,00 for {EQN}gammaM0.gif{/EQN} and 1,25 for {EQN}gammaM2.gif{/EQN} are recommended, but may be defined in the National Annex. If the recommended values are adopted, the above condition, for tension flanges less than 40mm, becomes

{EQN}AfnetAf81.gif{/EQN} for S275

{EQN}AfnetAf88.gif{/EQN} for S355

When {EQN}AfnetAf.gif{/EQN} is less than this limit, a reduced flange area {EQN}Af.gif{/EQN} may be assumed which satisfies the limit, that is the reduced flange area will be equal to {EQN}Afnet.gif{/EQN} divided by the limit value. Bolt holes in the tension zone of the web should be considered similarly but holes in the compression zone (both web and flange) may be ignored unless oversized or slotted.

{/DETAIL}

{/SUMMARY}

{TEST}

{TTITLE}Bolt holes{/TTITLE}

{QUESTION}

{QTITLE}Location of holes{/QTITLE}

{QTYPE}MC{/QTYPE}

{QTEXT}Bolt holes in which of the following parts of a cross-section require an additional check? {/QTEXT}

{ANSWER}

compression flange


{CHECK}Bolt holes in the compression flange can be ignored unless oversized or slotted holes are used.{/CHECK}


{UNCHECK}Bolt holes in the compression flange can be ignored unless oversized or slotted holes are used.{/UNCHECK}

{/ANSWER}

{ANSWER}

web (compression zone)


{CHECK}Bolt holes in the compression zone of the web can be ignored unless oversized or slotted holes are used.{/CHECK}


{UNCHECK}Bolt holes in the compression zone of the web can be ignored unless oversized or slotted holes are used.{/UNCHECK}

{/ANSWER}

{ANSWER}

tension flange


{CHECK}Yes – where bolt holes are located in the tension flange, additional checks may be necessary.{/CHECK}


{UNCHECK}No – where bolt holes are located in the tension flange, additional checks may be necessary.{/UNCHECK}

{/ANSWER}

{ANSWER}

web (tension zone)


{CHECK}Yes – where bolt holes are located in the tension zone of the web, additional checks may be necessary.{/CHECK}


{UNCHECK}Where bolt holes are located in the tension zone of the web, additional checks may be necessary.{/UNCHECK}

{/ANSWER}

{/QUESTION}

{QUESTION}

{QTITLE}Limiting hole size - S275{/QTITLE}

{QTYPE}N{/QTYPE}

{QTEXT}For S275 grade steel and a flange thickness less than 40mm what is the ratio of net to gross area of the tension flanges above which bolt holes can be neglected (to 2 decimal places) assuming the recommended values for the partial factors {EQN}gammaM0.gif{/EQN} and {EQN}gammaM2.gif{/EQN}?{/QTEXT}

{ANSWER}

{VARMIN}0.81{/VARMIN}

{VARMAX}0.81{/VARMAX}

{/ANSWER}

{FEEDBACK} Using values of 1,00 for {EQN}gammaM0.gif{/EQN} and 1,25 for {EQN}gammaM2.gif{/EQN} the general expression becomes

{EQN}AfnetAf81.gif{/EQN}

{/FEEDBACK}

{/QUESTION}

{QUESTION}

{QTITLE}Limiting hole size - S355{/QTITLE}

{QTYPE}N{/QTYPE}

{QTEXT}For S355 grade steel and a flange thickness less than 40mm what is the ratio of net to gross area of the tension flanges above which bolt holes can be neglected (to 2 decimal places) assuming the recommended values for the partial factors {EQN}gammaM0.gif{/EQN} and {EQN}gammaM2.gif{/EQN}?{/QTEXT}

{ANSWER}



{/ANSWER}

{FEEDBACK} Using values of 1,00 for {EQN}gammaM0.gif{/EQN} and 1,25 for {EQN}gammaM2.gif{/EQN} the general expression becomes

{EQN}AfnetAf88.gif{/EQN}

{/FEEDBACK}

{/QUESTION}

{/TEST}

{SUMMARY}

{SUMTITLE}Continuous structures{/SUMTITLE}

Continuous structures can be designed plastically, allowing the formation of successive plastic hinges, provided class 1 sections are used.

{DETAIL}

It should be noted here that for continuous (statically indeterminate) structures attainment of the design moment resistance at the point of maximum moment obtained from an elastic analysis will not normally lead to collapse (see figure below). Instead the cross-

section at that point will behave as a hinge - provided it has the requisite rotation capacity - and the pattern of moments within the structure will alter from the original elastic distribution as successive hinges form. Redistribution of moments enables the structure to withstand loads beyond that which produces the first hinge, until eventually sufficient hinges have formed to turn the structure into a mechanism. This is plastic design and requires a cross-section that can rotate whilst sustaining the plastic resistance moment i.e. a class 1 section is necessary.


{FIGURE}Figure 7. Load deflection curve for a statically indeterminate beam showing applied load {EQN}F.gif{/EQN} against vertical deflection {EQN}delta.gif{/EQN}. Actual behaviour shown in blue, behaviour according to simple plastic theory shown in red.

(a) Elastic (b) Elastic-plastic (c) Plastic

{EQN}Fy.gif{/EQN} - load at first yield; {EQN}F1.gif{/EQN} - load at first plastic hinge; {EQN}FC.gif{/EQN} - load at collapse{/FIGURE}

{/DETAIL}

{/SUMMARY}

{TEST}

{TTITLE}Collapse mechanisms{/TTITLE}

{QUESTION}

{QTITLE}Plastic hinge patterns for collapse{/QTITLE}

{QTYPE}N{/QTYPE}

{QTEXT}In a statically indeterminate system with three redundant degrees of freedom, how many plastic hinges are normally necessary to cause collapse?{/QTEXT}

{ANSWER}

{VARMIN}4{/VARMIN}

{VARMAX}4{/VARMAX}

{/ANSWER}

{/QUESTION}

{/TEST}

{/SECTION}

{SECTION}

{STITLE}Shear resistance{/STITLE}

{SUMMARY}

{SUMTITLE}Assumptions{/SUMTITLE}

Design for shear resistance, which is not normally a critical condition, is based on a simplified stress distribution and a material strength of {EQN}fyRt3.gif{/EQN}. The shear resistance is assumed to be provided only by that part of the cross-section in the same line as the direction of the shear force.

{PPT}Lecture11Shear res.pps{/PPT}

{DETAIL}

Bending governs the design of many steel beams but shear resistance can be significant for short beams with heavy concentrated loads. The following figure shows the pattern of shear stress in an I section assuming elastic behaviour. Almost all the shear force is carried by the web and since the variation in shear stress through the web is quite small it is sufficiently accurate for design to assume an average shear stress over the web.

Cross-section Distribution of shear stress {EQN}tau.gif{/EQN}

{IMAGE}L11I3a.jpg{/IMAGE} {IMAGE}L11I3b.jpg{/IMAGE}

{IMAGE}L11I3c.jpg{/IMAGE} {IMAGE}L11I3d.jpg{/IMAGE}

{FIGURE}Figure 8. Distribution of shear force in beams{/FIGURE}

Steel in shear yields at a stress approximately equal to {EQN}fyRt3.gif{/EQN}. Hence, the design value of the shear force {EQN}VSd.gif{/EQN} at each cross section is compared with the plastic shear resistance, {EQN}VplRd.gif{/EQN} of the shear area {EQN}Av.gif{/EQN}.



Shear areas for a range of section types are shown in the table below.

The above equation is valid for all webs, but webs which are not sufficiently stocky may fail prematurely by buckling in shear.

Shear buckling resistance must be checked if the web slenderness {EQN}dtw.gif{/EQN}exceeds the limits specified in {ECLINK}EC3 Part 1-5 paragraph 5.1(2){/ECLINK}. The limit depends on steel grade.

Rolled Load parallel to web

1,04 {EQN}h.gif{/EQN}{EQN}dtw.gif{/EQN}*

{IMAGE}Table21.wmf{/IMAGE}

I and H sections

Fabricated Load parallel to web

{EQN}h2tf.gif{/EQN}{EQN}tw.gif{/EQN} {IMAGE}Table21.wmf{/IMAGE}

Load parallel

to flanges

{EQN}h2tf.gif{/EQN}{EQN}tw.gif{/EQN}* {IMAGE}Table22.wmf{/IMAGE}

Rolled channel sections

Load parallel to web

1,04 {EQN}h.gif{/EQN}{EQN}dtw.gif{/EQN}*


Rolled angle sections

Load parallel

to longer

leg

{EQN}h.gif{/EQN}{EQN}t.gif{/EQN}*

Rolled rectangular hollow

Load parallel

to depth

{EQN}Ahbh.gif{/EQN} **


sections of uniform thickness

Load parallel

to breadth

{EQN}Ahbh.gif{/EQN} **

Circular hollow sections and tubes of uniform thickness

0,6 {EQN}A.gif{/EQN} ** {IMAGE}Table25.wmf{/IMAGE}

Plates and solid bars {EQN}A.gif{/EQN} ** {IMAGE}Table26.wmf{/IMAGE}

* This is an approximate formula. More accurate values of {EQN}Av.gif{/EQN} for rolled sections can be determined from the following expressions:

• for I and H sections: {EQN}Av.gif{/EQN} = {EQN}Avformula.gif{/EQN} • for channel sections: {EQN}Av.gif{/EQN} = {EQN}Avformula.gif{/EQN}

It is convenient to note that 1,04 /{EQN}Rt3.gif{/EQN} = 0,60 and so for a rolled I, H or channel section:

{EQN}VplRdformula.gif{/EQN}

** {EQN}A.gif{/EQN} is the total cross-sectional area

{FIGURE}Table 2 Shear Areas {EQN}Av.gif{/EQN} for Typical Sections{/FIGURE}

{/DETAIL}

{/SUMMARY}

{TEST}

{TTITLE}Shear resistance{/TTITLE}

{QUESTION}

{QTITLE}Shear area{/QTITLE}

{QTYPE}MC{/QTYPE}

{QTEXT}Which of the following parts of a cross section is mostly responsible for vertical shear resistance?{/QTEXT}

{ANSWER}

compression flange


{CHECK}The compression flange makes only a negligible contribution to shear resistance and is ignored.{/CHECK}


{UNCHECK}The compression flange makes only a negligible contribution to shear resistance and is ignored.{/UNCHECK}

{/ANSWER}

{ANSWER}

web


{CHECK}The web is the part of the cross section mostly responsible for vertical shear resistance.{/CHECK}


{UNCHECK}The web is the part of the cross section mostly responsible for vertical shear resistance.{/UNCHECK}

{/ANSWER}

{ANSWER}

tension flange


{CHECK}The tension flange makes only a negligible contribution to shear resistance and is ignored.{/CHECK}


{UNCHECK}The tension flange makes only a negligible contribution to shear resistance and is ignored.{/UNCHECK}

{/ANSWER}

{/QUESTION}

{/TEST}

{/SECTION}

{SECTION}

{STITLE}Moment resistance with high shear{/STITLE}

{SUMMARY}

{SUMTITLE}Effect of shear on moment resistance{/SUMTITLE}

The bending moment resistance of beams with high coexistent shear must be calculated based on a reduced cross-section.

{PPT}Lecture11M and V.pps{/PPT}

{DETAIL}

When the design shear force exceeds 50% of the plastic shear resistance, the design moment resistance of the cross-section is reduced to account for the interaction of shear and moment. It is assumed that under a combination of bending and shear stress steel yields in accordance with the interaction formula



The design plastic moment of a cross section carrying a coexistent significant shear is calculated using a reduced strength for the shear area. This reduced strength is dependent on the ratio of the shear load to shear capacity by the relationship,



and the reduced strength is {EQN}rhofy.gif{/EQN} for the shear area. For an I beam or H section bent about its major axis this leads to the reduced design plastic resistance moment (for Class 1 or 2 sections) in the presence of shear {EQN}MyRd.gif{/EQN}.


{ECLINK}EC3:Part 1-1:6.2.8(5) equation 6.30{/ECLINK}

{/DETAIL}

{/SUMMARY}

{TEST}

{TTITLE}Interaction of shear and bending{/TTITLE}

{QUESTION}

{QTITLE}'High' shear force definition{/QTITLE}

{QTYPE}N{/QTYPE}

{QTEXT}What is the maximum design shear force, as a proportion (between 0 and 1) of the plastic shear resistance of the section, above which it is necessary to allow for the effect on the moment resistance?{/QTEXT}

{ANSWER}



{/ANSWER}

{FEEDBACK}The effect of shear on bending resistance can be ignored if the design shear force is less than 50% of the plastic shear resistance of the cross-section.{/FEEDBACK}

{/QUESTION}

{/TEST}

{/SECTION}

{SECTION}

{STITLE}Biaxial bending{/STITLE}

{SUMMARY}

{SUMTITLE}Beams in biaxial bending{/SUMTITLE}

Beams in biaxial bending can be considered using an interaction curve

{DETAIL}

Beams bent about both axes of the cross-section have a plastic neutral axis inclined to the rectangular axes by an amount which depends on the ratio of the applied moments and the precise shape of the section. The following figure shows the interaction curve for full plasticity of an I section under biaxial loading.


{FIGURE}Figure 9. Interaction curve for full plasticity of an I-section under biaxial loading{/FIGURE}

The shape of the interaction can be expressed by


{ECLINK}EC3:Part 1-1:6.2.9.1(6) (6.41){/ECLINK}

{/DETAIL}

{/SUMMARY}

{/SECTION}

{SECTION}

{STITLE}Serviceability checks{/STITLE}

{SUMMARY}

{SUMTITLE}Deflection control{/SUMTITLE}

Beams must be checked to avoid excessive deflections or vibrations. These can cause damage to the building fabric and concern or discomfort to users.

{PPT}Lecture11Defs.pps{/PPT}

{DETAIL}

In addition to the strength checks outlined above, it is also necessary to check the performance of beams at the serviceability limit states. Deflections and vibrations of beams must be limited to avoid adverse affects to the appearance or effective use of the structure, discomfort to the occupants, or damage to the finishes and contents of a building. Acceptable limits for deflections should be agreed between the client, designer

and competent authorities. For guidance the table below gives recommended limiting values for vertical deflections.

Limits

Conditions {EQN}deltamax.gif{/EQN} {EQN}deltamin.gif{/EQN}

Roofs generally

L/200 L/250

Roofs frequently carrying personnel other than for maintenance

L/250 L/300

Floors generally

L/250 L/300

Floors and roofs supporting plaster or other brittle finish or non-flexible partitions

L/250 L/350

Floors supporting columns (unless the deflection has been included in the global analysis for the ultimate limit state)

L/400 L/500

Where δ max can impair the appearance

L/250 -

of the building

{FIGURE}Table 3 Recommended limiting values for vertical deflections{/FIGURE}

For structures open to the public it is important to ensure that oscillation and vibrations are not so great that they cause discomfort to users. Verification of the suitability of a design may be by means of a dynamic analysis but in many cases limiting the deflection is sufficient. For example, floors in dwellings and offices should have a lowest natural frequency not less than 3 cycles/second. This condition will be satisfied if the instantaneous total deflection (see table above) is less than 28mm. For floors in gymnasia or dance halls, the lowest natural frequency should not be less than 5 cycles/second - a deflection limit of 10mm would satisfy this limit.

Flat roofs (slopes less than 5 degrees) are vulnerable to ponding if the roof deflects such that pools of water may form. It is therefore necessary to check roof deflections carefully including allowances for construction inaccuracies, settlements of foundations, deflection of roof materials etc.

{/DETAIL}

{/SUMMARY}

{TEST}

{TTITLE}Deflection control{/TTITLE}

{QUESTION}

{QTITLE}Effect of excessive deflections{/QTITLE}

{QTYPE}MC{/QTYPE}

{QTEXT}Excessive deflections may cause damage to or impair which of the following {/QTEXT}

{ANSWER}

Appearance


{CHECK}Yes – appearance can be impaired by excessive deflections{/CHECK}


{UNCHECK}Appearance can be impaired by excessive deflections {/UNCHECK}

{/ANSWER}

{ANSWER}

Damage to finishes


{CHECK}Yes –finishes can be damaged by excessive deflections{/CHECK}


{UNCHECK}Finishes can be damaged by excessive deflections {/UNCHECK}

{/ANSWER}

{ANSWER}

Effective use of the structure


{CHECK}Yes – effective use of the structure can be impaired by excessive deflections{/CHECK}


{UNCHECK}Effective use of the structure can be impaired by excessive deflections.{/UNCHECK}

{/ANSWER}

{ANSWER}

bending strength


{CHECK}No – strength is a separate consideration from deflection{/CHECK}


{UNCHECK}Strength is a separate consideration from deflection{/UNCHECK}

{/ANSWER}

{ANSWER}

shear strength


{CHECK}No – strength is a separate consideration from deflection{/CHECK}


{UNCHECK}Strength is a separate consideration from deflection{/UNCHECK}

{/ANSWER}

{/QUESTION}

{QUESTION}

{QTITLE}Deflection limits{/QTITLE}

{QTYPE}MC{/QTYPE}

{QTEXT}Maximum deflections are normally limited to a proportion of which of the following {/QTEXT}

{ANSWER}

beam span


{CHECK}Yes - maximum deflections are normally limited to a proportion of span{/CHECK}


{UNCHECK}Maximum deflections are normally limited to a proportion of span{/UNCHECK}

{/ANSWER}

{ANSWER}

Beam depth


{CHECK}No - beam depth will influence actual deflections but not the allowable limit.{/CHECK}


{UNCHECK}Beam depth will influence actual deflections but not the allowable limit.{/UNCHECK}

{/ANSWER}

{/QUESTION}

{/TEST}

{/SECTION}

{SECTION}

{STITLE}Concluding summary{/STITLE}

{SUMMARY}

• A variety of section shapes are available for beams; choice depends on load and span.

• Beams are generally designed on the basis of bending moment resistance. • Stiffness and its influence on deflection under serviceability loads is an important

consideration. • Beams which are prevented from lateral movement are termed restrained. • Moment resistance is dependent on section classification. • The effect of co-existent shear forces on the moment resistance can be ignored if

they are less than 50% of the plastic shear resistance.

{/SUMMARY}

{/SECTION}

{/LECTURE}

Documents

NFATEC – L11 – Restrained beams (25/02/2004)library.tee.gr/digital/m2247/m2247_l11_eng.pdf · Beam Type Span Range (m) Notes 0. ... Castellated beam{/FIGURE} {IMAGE}Plate girder