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STATICALLY INDETERMINATE BEAMS

1. INTRODUCTION

Most of the structures encountered in real-life are Statically Indeterminate.

Statically Indeterminate Beams: No. of Reactions > No. of Eqns. of Equilibrium

Degree of Static Indeterminacy = No. of Reactions in excess of the No. of Eqns of Equilibrium.

Static Redundants = excess reactions; must be selected for each particular case.

Assumption throughout this chapter is that the beams are made of Linearly Elastic Materials.

2. TYPES OF STATICALLY INDETERMINATE BEAMS

- Propped Cantilever Beam

- Fixed – End Beam

- Continuous Beam

(more than one span)

There are 4 ways of solving these types of problems.

1. Use of the deflection curve

2. Moment – Area Method

3. Superposition (Flexibility Method)

4. Indeterminate Beams Tables (handout)

We will examine No. 1 & 3, above.

3. ANALYSIS BY THE DIFFERENTIAL EQUATIONS OF THE DEFLECTION CURVE

1. pick redundant reaction

2. express other reactions in terms of the redundant reaction

3. write diff. eqn. of the deflection curve

4. integrate to obtain general solution

5. apply B.C. to obtain constants of integration & the redundant reaction

6. solve for the remaining reactions using equations or equilibrium

This method is useful for:

- simple loading conditions

- beams of only one span (not good method for continuous beam)

EXAMPLE No. 1

GIVEN:

The beam shown.

FIND:

Reactions at supports using the deflection curve.

SOLn:

MOMENT – AREA METHOD (just mention)

1. pick redundant reaction(s)

2. remove redundant reaction(s) to leave a statically determinate beam

3. apply loads on released structure

4. draw M / EI diagram for these loads

5. apply redundant reactions as loads

6. draw M / EI diagram for redundant reactions

7. apply moment – area theorems to find redundant...

1. INTRODUCTION

Most of the structures encountered in real-life are Statically Indeterminate.

Statically Indeterminate Beams: No. of Reactions > No. of Eqns. of Equilibrium

Degree of Static Indeterminacy = No. of Reactions in excess of the No. of Eqns of Equilibrium.

Static Redundants = excess reactions; must be selected for each particular case.

Assumption throughout this chapter is that the beams are made of Linearly Elastic Materials.

2. TYPES OF STATICALLY INDETERMINATE BEAMS

- Propped Cantilever Beam

- Fixed – End Beam

- Continuous Beam

(more than one span)

There are 4 ways of solving these types of problems.

1. Use of the deflection curve

2. Moment – Area Method

3. Superposition (Flexibility Method)

4. Indeterminate Beams Tables (handout)

We will examine No. 1 & 3, above.

3. ANALYSIS BY THE DIFFERENTIAL EQUATIONS OF THE DEFLECTION CURVE

1. pick redundant reaction

2. express other reactions in terms of the redundant reaction

3. write diff. eqn. of the deflection curve

4. integrate to obtain general solution

5. apply B.C. to obtain constants of integration & the redundant reaction

6. solve for the remaining reactions using equations or equilibrium

This method is useful for:

- simple loading conditions

- beams of only one span (not good method for continuous beam)

EXAMPLE No. 1

GIVEN:

The beam shown.

FIND:

Reactions at supports using the deflection curve.

SOLn:

MOMENT – AREA METHOD (just mention)

1. pick redundant reaction(s)

2. remove redundant reaction(s) to leave a statically determinate beam

3. apply loads on released structure

4. draw M / EI diagram for these loads

5. apply redundant reactions as loads

6. draw M / EI diagram for redundant reactions

7. apply moment – area theorems to find redundant...

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