Analyzing Statically Indeterminate Structures: A Step-by-Step Force Method, Summaries of Decision Making

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Analysis of Statically Indeterminate

Structures Using the Force Method

Steven Vukazich

San Jose State University

Statically Indeterminate Structures

At the beginning of the course, we learned that a stable

structure that contains more unknowns than independent

equations of equilibrium is Statically Indeterminate.

• Redundancy (several members

must fail for the structure to

become unstable);

• Often maximum stresses is

certain members are reduced;

• Usually deflections are

reduced.

Advantages

Disadvantages

• Connections are often more

expensive;

• Finding forces and deflections

using hand analysis is much

more complicated.

Force Method of Analysis

E

B

A

C

w

P

Consider the beam

FBD

E

B

A

D

w

P

A

x

A

y

M

A

D

y

X = 5

3 n = 3(1) = 3

Beam is stable

Statically Indeterminate

to the 2

nd

degree

EI

D

C

C

y

Define Primary Structure and Redundants

• 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.

E

B

A

D

Primary Structure

Redundants

M

A

E

A B D

D

y

E

B

A

D

M

Q

C

C

C y C y C y Q

C

Define and Solve the Redundant Problems

• 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;

E

B

A

y

x

EI

D

C

y

E

A B

y

x

EI

D

'

C

"#

"#

C

Redundant

Problem 1

"#

'

"#

Define and Solve the Redundant Problems

E

B

A

y

x

#"

EI

D

D

y

B E

A

y

x

#"

EI

D

#"

'

#"

C

""

""

C

Redundant

Problem 2

""

'

""

The Force

Method is

Based on the

Principle of

Superposition

Indeterminate

Problem

Primary

Problem

Redundant

Problem 1

Redundant

Problem 2

E

B

A

C

w

P

EI

D

D

B

A

C

w

P

y

x

"

EI

E

E

B

A

y

x

#"

EI

D

D

y

C

""

E

B

A

y

x

EI

C D

"#

C

y

y

x

Example Problem

A

B

P

EI

C

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

y

x

A

x

A

y

M

A

C

y

FBD

X = 4

3 n = 3(1) = 3

Beam is stable

Statically Indeterminate

to the 1

st

degree

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

P

y

x

EI

/

𝜃

= −

𝑃𝐿

16 𝐸𝐼

From

Tabulated

Solutions

Counter-clockwise

rotations positive

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

B

A

C

y

x

EI

//

M

A

B

A

C

y

x

EI

//

𝛼

= −

𝐿

3 𝐸𝐼

From Tabulated

Solutions

L

L

A

B

P

EI

C

y

x

A

x

A

y

M

A

C

y

Free Body Diagram

𝑀

= −

3

16

𝑃𝐿

A

B

P

EI

C

y x A x A y

C

y

3

16

𝑃𝐿

Can now use

equilibrium

equations to find

the remaining

three unknowns

Find Remaining Unknowns

/

=

'

A

x

A

B

P

EI

C

y x A x A y

C

y

3

16

𝑃𝐿

Can now use

equilibrium

equations to find

the remaining

three unknowns

𝐶

=

5

16

𝑃

𝐴

=

11

16

𝑃

V

M 0

Superposition of Primary and Redundant Problems

A

C

P

B

P

B

B

Indeterminate

Problem

Primary

Problem

Redundant

Problem

C

C

A A