1

Structural Design and Inspection-Deflection and Slope of Beams

By

Dr. Mahdi Damghani

2016-2017

2

Suggested Readings

Reference 1Chapter 16

3

Objective

• To obtain slope and deflection of beam and frame structures using slope-deflection method

4

Introduction

• Structural analysis method for beams and frames introduced in 1914 by George A. Maney

• This method was later replaced by moment distribution method which is more advanced and useful (students are encouraged) to study this separately

5

Slope-Deflection Method

• Sign convention:• Moments, slopes,

displacements, shear are all in positive direction as shown• Axial forces are ignored

xSMdxvdEI ABAB2

2

21

32

1

2

62

2

CxCxSxMEIv

CxSxMdxdvEI

ABAB

ABAB

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Slope-Deflection Method

AA vvdxdvx ,0@

AAABAB

AABAB

EIvxEIxSxMEIv

EIxSxMdxdvEI

62

232

2

BB vvdxdvLx ,@

AAABABB

AABABB

EIvLEILSLMEIv

EILSLMEI

62

232

2

Solving for MAB and SAB

BABAAB vv

LLEIM 322

BABAAB vv

LLEIS 262

BABABA vv

LLEIM 322

BABABA vv

LLEIS 262

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Slope-Deflection Method

• Last 4 equations obtained in previous slide are called slope-deflection equations

• They establish force-displacement relationship

• This method can find exact solution to indeterminate structures

8

Slope-Deflection Method

• The beam we considered so far did not have any external loading from A to B

• In the presence of mid-span loading (common engineering problems) the equations become:

FABBABAAB Mvv

LLEIM

322 FABBABAAB Svv

LLEIS

262

FBABABABA Mvv

LLEIM

322 FBABABABA Svv

LLEIS

262

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Fixed End Moment/Shear

• MABF, MBA

F are fixed end moments at nodes A and B, respectively.

• Moments at two ends of beam when beam is clamped at both ends under external loading (see next slides)

• SABF, SBA

F are fixed end shears at nodes A and B, respectively.

• Shears at two ends of beam when beam is clamped at both ends under external loading (see next slides)

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Fixed End Moment/Shear

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Fixed End Moment/Shear

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Example

• Find support reactions.

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Solution

• This beam has 2 degrees of indeterminacy

1- Assume all beams are fixed & calculate FEM

2- Establish Slope-Deflection

equations

3- Enforce boundary

conditions & equilibrium

conditions at joints

4- Solve simultaneous

equations to get slopes/deflection

s

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Solution

kNmMM FBA

FAB 75.0

80.16 kNmMM F

CBFBC 25.1

80.110 kNmMM F

DCFCD 00.1

120.112 2

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Solution

Fijjijiij Mvv

LLEIM

322

Fjijijiji Mvv

LLEIM

322

vi=0 and vj=0 for all cases

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Solution

• Equilibrium moments at the joints

MBA MBCMCB MCD

0ABM

0

0

BCBA

B

MM

M

0

0

CDCB

C

MM

M 0DCM

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Solution

• Substitution into slope deflection equations gives 4 equations and 4 unknown slopes.

• By simultaneously solving the equations

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Solution

• Simply operation of substitution:

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Solution

• Now support reactions can easily be calculated as

15.485.16

85.10.115.15.06

BA

AB

R

R

6 10

25.575.410

75.40.1

4.115.15.010

CB

BC

R

R

12

6.44.712

4.70.1

4.15.012

DC

CD

R

R

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Solution

1.85 4.15 4.75 5.25 7.4 4.6

1.85 8.9 12.65 4.60

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Solution

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Example 2

• Obtain moment reaction at the clamped support B for 6m long beam if support A settles down by 5mm. EI=17×1012 Nmm2

A

4kN/m

B

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Solution

Fijjijiij Mvv

LLEIM

322 Fjijijiji Mvv

LLEIM

322

A

4kN/m5mm

B

F

ABBABAAB MvvLL

EIM 322

1240000005.030220

2LLL

EIM AAB

radLL

EIA

4

3

1033.92

0015.0244000

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Solution

A

4kN/m5mm

B

Fijjijiij Mvv

LLEIM

322 Fjijijiji Mvv

LLEIM

322

radA41033.9

F

BABAABBA MvvLL

EIM 322

1240000005.031033.902 2

4 LLL

EIM BA

mNM BA .3.18705

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Case study

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Case study

t=3.552 mmb=7.792 mm

Fijjijiij Mvv

LLEIM

322

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Case study

Rotation and displacement can be obtained from the

FEM

Wire in composite beam

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Case study

Rotation and displacement can be obtained from the

FEM

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Q1

• Determine the support reactions in the beam shown below.

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Q2

• Calculate the support reactions in the beam shown below.

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Q3

• Determine the end moments in the members of the portal frame shown. The second moment of area of the vertical members is 2.5I while that of the horizontal members is I.

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Q4

• Analyze two span continuous beam ABC by slope deflection method. Then draw Bending moment & Shear force diagram. Take EI constant.