216636 4 question 2 worked example

Analysis & Reinforced concrete beam design

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Figure 21: One-way slab (supported on two opposite edges)

continuous beams Continuous beams (they have more than one span) are more common instructures and buildings. Their end supports may be fixed or hinged.

Continuous supports differ from hinges because the beam over thesesupports resist some moment, whereas hinges cannot resist moment.

Figure 22(a): Simply supported beam

methods of analysis Some of the well known methods of analysis of continuous beams andone-way slabs are:

moment distributiondirect stiffness matrix method

The analysis of structural members will be covered in detail in the Theoryand Design of Structures course which is offered in the third year of thisprogram.

In this topic we will analyse the continuous beams and continuousone-way slabs using a simplified method which is given in your textbookand in the Australian Standard AS3600, Chapter 1 of HB2.2. I willexplain how to refer and use the coefficients from Figures 9.3 and 9.4 ofyour textbook to calculate the bending moment (BM) and shear force fortwo or more span continuous beams and one-way slabs.


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Figure 22(b): Continuous beam with simple end support

Using the simplified method for bending momentand shear force diagramsdata required for analysis If you want to analyse a reinforced concrete continuous beam/one-way

slab, you must know the:

type and magnitude of load acting on the structural elementsspans of the beam/slabend conditions.

conditions required for thesimplified method

If you follow the simplified method for bending moment and shear forcecalculations then the spans and the loads acting on the beam must satisfyseveral conditions. These conditions are given in your textbook. Pleaseread the following pages before you start to analyse any continuousbeam/one-way slab. The same information is also given in SAA HB2.2.

Textbook Beletich & Uno 2003Ch. 9, Sections 9.19.3

Australian Standards SAA HB 2.2Ch. 1, Section 7.2

Equations for design bending momentand design shear forceFrom the textbook, we have equations which can be used to calculatedesign bending moment and design shear force.

For the design bending moment:

2* tcoefficien nd LFBMM

where

supports.offacesinsidebetweenspancleartheisloaddesignddistributeuniformlytheis

momentbendingdesigntheis

n

d

*

LF

M


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For the design shear force V*

nd LFSFV tcoefficien*

where Fd and Ln are the same as explained in the equation above.

BM and SF coefficientsFigure 9.3 from the textbook gives you the coefficients for SF and BM fortwo-span continuous beams or slabs. The BM coefficients vary accordingto the end-support conditions. The SF coefficients remain the same fordifferent end-support conditions.

Figure 9.4 from the text gives you the coefficients for SF and BM forthree or more spans. Again the BM coefficients vary according to the end-support conditions and the SF coefficients remain the same for differentend-support conditions.

Six required conditionsTo use the coefficients, six conditions must be satisfied as outlined in thetextbook.

Now let us analyse a continuous beam using the simplified method, and topractice using Figure 9.3 and Equations 9.1 and 9.2 from your textbookfor bending moment and shear force calculations.

Example 21Problem Draw the shear force and bending moment diagrams for the twospan

continuous beam ABC as shown in Figure 23 carrying a live load of 9 kN/m and a dead load of 6 kN/m. AB and BC have the same span of7 m. The beam has simple end supports and a uniform cross section.

Figure 23: Two-span continuous beam


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Example 21 continuedSolution Check that the conditions given in the textbook are satisfied

(a) ratio of longer to shorter span = 7:7= 1.0

span ratio


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Example 21 continued

Span AB coefficient clear span

nL (m) (kNm)textbook)from9.1(Eqn)(**tcoefficienBM* 2nd LFM

at support A 0 7 20 20.7 7 0

mid span AB

111 7 21 20.7 7 92.2

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at support B

91 7 21 20.7 7 112.7

9

Span BC

at support B91 7 21 20.7 7 112.7

9

mid span BC

111 7 21 20.7 7 92.2

11

at support C 0 7 20 20.7 7 0

Table 22: Bending moment diagram calculations

Draw the SFD and BMD using the values from Tables 21 and 22 as shown in Figure 24.


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Example 21 continued

Beam

A 7 m B 7 m C

112.7

92.2 92.2

72.5

20.7

20.7

83.3

83.3

20.7

20.7

72.5

Units in kN

Units in kNmBMD (Bending Moment Diagram)

SFD (Shear Force Diagram)

D

Figure 24: Bending moment and shear force diagrams

End of Example 21

Workbook activity 21 As shown in Figure 25, a 200 mm thick reinforced concrete platform (slab) is supported by beams with a cross sectional dimension(width depth) of 400 600 mm. The beams are supported by columns atthe ends. The density of the concrete is 2500 kg/m3. This platform is partof a workshop.

1. Draw the SF and BM diagrams of the reinforced concrete slab.2. Draw the BM diagram of the beam BD (assume fixed supports).

hint Refer to Section 3, Chapter 5 of HB2.2, for the live load intensity on theslab. When calculating the dead load, you can refer to Topic 1 if required.For load calculations the slab may be treated as a beam of 1m width.Then find the design load.

The 200 mm thick slab is considered to be a two-span one-waycontinuous slab. Use the correct SF and BM coefficients from Figure 9.3in the textbook.


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The beam BD is not a continuous beam. Since the slab is not supported onits four sides the slab area is divided into rectangular shapes as shown inFigure 26 for load calculations on the beam.

Figure 25: Reinforced concrete platform

Figure 26: Overhead view of reinforced concrete platform

BM coefficients for a beam carrying UDL with fixed support is given inFigure 27.


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Figure 27: BM coefficients for a fixed end beam

Solution to Activity 21 1. Design load = 13.4 kN/m for a metre width strip. The BM and SF values are given in Table 23. Check your answers and then

draw the SFD and BMD.

Span AB (slab)S BM (kNm) SF (kN)

at support A 14.0 33.5

mid span 30.4 9.6

at support B 37.2 38.5

Table 23: BM and SF values

The slab has two equal spans. Therefore the BM and SF on the span BC also hasthe same values as shown in Table 23.

2. Design load = 77.0 kN/m The BM values are given in Table 24. Check your answers and then draw the

BMD.

Beam Span BD B.M (kNm)

at support B 231

mid span 116

at support B 231

Table 24: BM values

216636 4 question 2 worked example

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