Chapter 10 Force Method

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Analysis of Statically Indeterminate Structures

Iqbal Marie 2018-2019

Structural Analysis- Text Book by: R. C. HIBBELER

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Structures

Determinate Indeterminate

Serious effect No effect due due settlement settlement

No effect due to Serious effect due to Rise in rise in temperature temperature

Not economical economical

Deflection

Formation of plastic hinges Formation of results unstable plastic hinges structure results stable structure

https://www.youtube.com/watch?v=Ff0GsQpw0.Xc

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There are two different Methods of analysis Force method • known as consistent deformation, unit load method, flexibility method • The primary unknowns are forces

Displacement method • Known as stiffness method • The primary unknowns are displacements

The deflection or slope at any point on a structure as a result of a number of forces, including the reactions, is equal to the algebraic sum of the deflections or slopes at this particular point as a result of these loads acting individually

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10.1 – 10-3 Force Method of Analysis: Beams 10.2 Analysis Procedure

Indeterminate to the first degree ( 4 unknowns)

• Three equilibrium equations and one compatibility equation is needed • Choosing one of the support reaction as a redundant (say

By) • The structure become statically determinate and stable ( primary structure)

• Calculate Downward displacement B at B (load action) • Apply unit load at B and calculate BB (upward deflection )- correction structure BB

• Apply the Compatibility equation: actual beam = B+ By BB Correction structure to find By • Now the structure is statically determinate and the

equilibrium equations can be used to find the other CB reactions • Final moment or reaction can be calculated as:

By actual beam = 0 for no settlement or spring By

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Indeterminate to the 2nd degree

• 2 Compatibility equations are needed • Select two redundant reaction B & C

• Determine displacement B &C caused by loads • BB & BC Deflection per unit force at B are determined • CC & CB Deflection per unit force at C are determined • Compatibility equations are applied to find reactions at B and C

• 0 = B+ By BB + Cy BC • 0 = C+ By CB + Cy CC • Then find the other reaction by applying the equilibrium equation

 CB By Cy BB

C By y

BC CC

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10.3 Maxwell’s Theorem

BB CB

BC = CB

BC CC

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Draw bending moment diagrams for the beam

300

3 B=1/6(6+2x12)(6x-300)/EI= -999kN.m /EI 12 BB BB =1/3(12x12x12) /EI= 576 /EI compatibility equation:

0 = B+ By  BB -999+ By (576) = 0 By = 15.6 kN Apply equilibrium equations 6

Ay = 34.4 kN MA= 112.8 kNm

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Draw shear and bending moment diagrams for the beam. The support at B settles 1.5 in ( 40mm). E= 29(103) ksi, I = 750 in4

BB

120

M 6

B= 1/3(180x6x12) + 1/3(120x12x24)+1/6(2x180x6 + 180x12 + 120x6 +2x120x12)12 = 31680/EI m

BB = 1/3( 12x12x48) = 2304/EI

BB

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Determine the support reactions on the frame shown EI is Constant

BB

 20 BB

4 B =1/3(25x4x5) + 0 = 166.7/EI  =1/3(4x4x5 )+1/3(4x4x4) = 48/EI 4 BB 25kN.m

M m

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Draw bending moment diagram

aa= 1/3(5x5x5)x2 +2 a= -1/3(5x50x5)- (1/6)(2x5x5+5x8+ 8x5+ 2x8x8)x5= 1/6(2x50x5+50x8+55x5+2x8x55)x5 – 513.33/EI (1/6)(5+2x8)x55x5 =- 3091.67/EI

actual beam = a+ Ax aa A Ax x

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a =[(1/3)3 x3x 12 R +[(1/3)x1.5x3x6 +(1/3)x3x1.5x6 +(1/6)(2x3*1.5+3x2.25+1.5x1.5+2*2.25x1.5 )(3)]x10 =[36R +247.5]/EI

+ 2.25

4 a =[0.25(16x4)(4)/2+(6- R)/6(16)(4) = [32+(6- R)*10.66]/EI

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https://www.chegg.com/homework-help/using-tables-e1-e2-deflections-slopes-beams-apply-force-meth- chapter-7.6-problem-40p-solution-9781118136348-exc

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10.6 Force Method of Analysis: Truss a. Externally indeterminate b. Internally indeterminate The degree of indeterminacy of a truss : b + r >2j.

The force method is quite suitable for analyzing trusses that are

statically indeterminate to the first or second degree member L A N n NnL/EA nnL/EA Nf

Procedure for analysis: a. Chose the redundant ( support reaction or redundant member)- [X] b. Convert the indeterminate structure to determinate and stable primary structure by removing the redundant c. Calculate the reactions ( R0)and (N) forces in each truss member of the primary structure due to external loads d. Apply a unit load in the direction of redundant reaction or a pair of unit loads in direction of redundant member ( always assumed as tensile) e. Calculate the (n) forces in the truss members due to unit load f. Tabulate the results and calculate:   g. Calculate the redundant as: +x = 0 h. Calculate the final reactions and the final member forces :

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 