Statically Indeterminate Frame ExampleStevenVukazich

SanJoseStateUniversity

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

Construct Internal Force

Diagrams (if necessary)

10 ft

30 kB10 k

5 ft 5 ft

A

C

D

Example Problem

For the indeterminate frame subjected to the point loads shown, find the support reactions and draw the bending moment diagram for the frame. EIis the same for both the horizontal and vertical members.

EI

EI

FBD of the Frame

10 ft

30 kB10 k

5 ft 5 ft

C

D

EI

EI

Ax

Ay

MA

Dy

X = 4

3n = 3(1) = 3

Frame 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.

Primary Structure Redundant

B

A

CMA

D

B

A

CD

Dy

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.

10 ft

30 kB10 k

5 ft 5 ft

A

C

D

EI

EI Δ"

Need to find Δ"Use the Principle of Virtual Work

Solve the Primary Problem

10 ft

30 kB10 k

5 ft 5 ft

A

C

D

EI

EI Δ"

Real System

Need to construct the Mpdiagram

Solve the Primary Problem

10 ft

B

5 ft 5 ft

A

C

D

EI

EI

Virtual System to measure Δ"

Need to construct the MQdiagram

1

FBD of the Primary Problem

#𝑀%

�

�

= 0+

#𝐹+

�

�

= 0+

#𝐹,

�

�

= 0+

MA = 250 k-ft

Ax = – 10 k

Ay = 30 k

10 ft

30 kB10 k

5 ft 5 ft

C

D

EI

EI

C

Ax

Ay

MA

B

A

D

C

150 k-ft

0

250 k-ft

0

MP Diagram for the Primary Problem

Moment diagram is drawn on the compression side of the member

FBD of the Virtual System

#𝑀%

�

�

= 0+

#𝐹+

�

�

= 0+

#𝐹,

�

�

= 0+

MA = 10 ft

Ax = 0

Ay = –1

10 ft

1

B10 k

5 ft 5 ft

C

D

EI

EI

C

Ax

Ay

MA

B

A

D

C

10 ft

10 ft

0

MQ Diagram for the Primary Problem

Moment diagram is drawn on the compression side of the member

10 ft5 ft

1 . ∆"= 1𝐸𝐼3 𝑀4𝑀5

6

7𝑑𝑥

Solve the Primary Problem

BA C10 ft 5 ftB

10 ft 10 ft5 ft

150 k-ft250 k-ft

150 k-ft

0

−16

5ft + 2 10ft 150k−ft 5ft

−12𝑀A 𝑀B + 𝑀C 𝐿

−12

10ft 250k−ft + 150k−ft 10ft

−16𝑀A + 2𝑀E 𝑀B𝐿

M1

M3M4

M1M2

∆"= −23,125k−ft3

𝐸𝐼

M3

L L

–20,000 k-ft3 –3,125 k-ft3

Define and Solve the Redundant Problem

• 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;

∆""= 𝐷,𝛿""

Redundant Problem

10 ft

B

5 ft 5 ft

A

C D

EI

EIΔ""

Dy

Define and Solve the Redundant Problem

• 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;

∆""= 𝐷,𝛿""

Flexibility Coefficient

10 ft

B

5 ft 5 ft

A

C D

EI

EI𝛿""

1

Need to find 𝛿""Use the Principle of Virtual Work

Solve the Flexibility Coefficient Problem

Real System

Need to construct the Mpdiagram

10 ft

B

5 ft 5 ft

A

C D

EI

EI𝛿""

1

Solve the Flexibility Coefficient Problem

Virtual System to measure 𝛿""

Need to construct the MQdiagram

10 ft

B

5 ft 5 ft

A

C D

EI

EI𝛿""

1 Note that for the flexibility coefficient problem the real and virtual systems are identical

FBD of the Flexibility Coefficient Problem

#𝑀%

�

�

= 0+

#𝐹+

�

�

= 0+

#𝐹,

�

�

= 0+

MA = 10 ft

Ax = 0

Ay = –1

10 ft

1

B10 k

5 ft 5 ft

C

D

EI

EI

C

Ax

Ay

MA

B

A

D

C

10 ft

10 ft

0

MP Diagram for the Flexibility Coefficient Problem

Moment diagram is drawn on the compression side of the member

10 ft

B

A

D

C

10 ft

10 ft

0

MQ Diagram for the Flexibility Coefficient Problem

Moment diagram is drawn on the compression side of the member

10 ft

1 . δ"" = 1𝐸𝐼3 𝑀4𝑀5

6

7𝑑𝑥

Solve the Flexibility Coefficient Problem

BA D10 ft 10 ftB

10 ft 10 ft

0

M1

M3

M1

𝛿"" =1333ft3

𝐸𝐼

M3

L L

1000 ft3 333.333 ft3

10 ft10 ft

13𝑀A𝑀B𝐿𝑀A𝑀B𝐿

10ft 10ft 10ft 13

10ft 10ft 10ft

∆""= 𝐷,1333ft3

𝐸𝐼

∆""= 𝐷,𝛿""

Compatibility Equation at Point D

∆" + ∆""= 0

Compatibility at Point D

Compatibility Equation in terms of Redundant and Flexibility Coefficient

Δ" + 𝐷,𝛿"" = 0

−23,125k−ft3

𝐸𝐼+ 𝐷,

1333ft3

𝐸𝐼= 0

𝐷, =23,125k−ft3

𝐸𝐼𝐸𝐼

1333ft3𝐷, = 17.34k

Solve for Dy

FBD of the Frame

10 ft

30 kB10 k

5 ft 5 ft

C

D

EI

EI

Ax

Ay

MA

17.34 k

𝐷, = 17.34k

#𝑀%

�

�

= 0+

#𝐹+

�

�

= 0+

#𝐹,

�

�

= 0+

MA = 76.6 k-ft

Ax = – 10 k

Ay = 12.66 k

B

A

D

C

76.6 k-ft

23.4 k-ft0

Moment Diagram for the Frame

Moment diagram is drawn on the compression side of the member

86.7 k-ft

B

A

DC

0

250 k-ft

0

Moment Diagrams for the Primary and Redundant Problems

150 k-ft

B DC

173.4 k-ft

173.4 k-ft

86.7 k-ft

Moment diagram is drawn on the compression side of the member

0

A

++ +

For horizontal member BDE

Choose sign convention for internal forces for bothhorizontal and vertical members

For vertical member ABC++ +