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Statically Indeterminate Structures Force Method Example Steven Vukazich San Jose State University

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Statically Indeterminate Structures Force Method Example

Steven VukazichSan Jose State University

Steps in Solving an Indeterminate Structure using the Force Method

Determine degree of Indeterminacy Let n =degree of indeterminacy

(i.e. the structure is indeterminate to the nth degree)

Define Primary Structure and the n Redundants

Define the Primary Problem

Solve for the nRelevant

Deflections in Primary Problem

Define the nRedundant Problems

Solve for the nRelevant Deflections in each Redundant

Problem

Write the nCompatibility Equations at

Relevant Points

Solve the nCompatibility

Equations to find the n Redundants

Use the Equations of Equilibrium to

solve for the remaining unknowns

Chapter 3

Chapters 3,4,5 then 7 or 8

Construct Internal Force

Diagrams (if necessary)

Chapter 3

Chapters 3,4,5 then 7 or 8

Chapters 3,4,5

Example Problem

AB

P

EI C

!2

!2

y

x

For the indeterminate beam subject to the point load, P, find the support reactions at A and C. EI is constant.

A B

P

EI C

!2

!2

y

xAx

Ay

MA

Cy

FBD

X = 4

3n = 3(1) = 3

Beam is stable

Statically Indeterminate to the 1st degree

Define Primary Structure and Redundant• Remove all applied loads from the actual structure;• Remove support reactions or internal forces to define a primary structure;• Removed reactions or internal forces are called redundants;• Same number of redundants as degree of indeterminacy• Primary structure must be stable and statically determinate;• Primary structure is not unique – there are several choices.

BAC

Primary Structure Redundant

MA

BA CCy

Define and Solve the Primary Problem

• Apply all loads on actual structure to the primary structure;• Define a reference coordinate system;• Calculate relevant deflections at points where redundants were

removed.

B

A C

Py

x

EI!"

#2

#2

!" = − '#(16+,

From Tabulated Solutions

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Tabulated Solutions

–+

+"#

Define and Solve the Redundant Problem• There are the same number of redundant problems as degrees of indeterminacy;• Define a reference coordinate system;• Apply only one redundant to the primary structure;• Write the redundant deflection in terms of the flexibility coefficient and the

redundant for each redundant problem.• Calculate the flexibility coefficient associated with the relevant deflections for

each redundant problem;

!"" = $"%""Redundant Problem

BA C

y

x

EI!""MA

BA C

y

x

EI%""1

%"" = − '3)*

From Tabulated Solutions

L

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Tabulated Solutions

–+

+"#

Compatibility Equation at Point A

!" + !"" = 0Compatibility at Point A

Compatibility Equation in terms of Redundant and Flexibility Coefficient

!" + &"'"" = 0

− )*+16./ +&" − *

3./ = 0

&" =)*+16./ −3./*

&" = − 316)*

Solve for MA

A B

P

EI C

!2

!2

y

xAx

Ay

MA

Cy

Free Body Diagram

#$ = − 316*!

A B

P

EI C

!2

!2

y

xAx

Ay Cy

316*! Can now use

equilibrium equations to find the remaining three unknowns

Find Remaining Unknowns

!"# = 0+

!'( = 0+

!') = 0+

Ax = 0

A B

P

EI C

,2

,2

y

xAx

Ay Cy

3161, Can now use

equilibrium equations to find the remaining three unknowns

2) =5161

4) =11161

+

+V

M0

Draw V and M Diagrams of the Beam

A B

P

EI C

"2

"2

y

x

316'"

516'

1116'

1116 '

− 516 '

*+ = − 316'" +

1132 '" =

532'"

*+ − *- =1116 '

"2

− 3'"16

532 '"

V

M 0

Superposition of Primary and Redundant Problems

A

C

P

B!2

!2

316 &!

516 &

1116 &

1116 &

− 316&!

532 &!

P

12 &

14 &!

316 &

− 332&!

− 12&

12&

12 &

316 &

316 &

316 &!

− 316&!

B B

0

IndeterminateProblem

PrimaryProblem

RedundantProblem= +

C CA A

0

− 516&

0