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Indeterminate Beams

Unlike determinate beams, indeterminate beams have more unknown reactions than available static equilibrium equations. This means that while we still use methods like the Double Integration Method, Moment-Area Theorems, or the Conjugate Beam Method to compute slopes and deflections, additional compatibility conditions must be introduced to solve for the unknowns.

In statically indeterminate beams, deflections or rotations at certain points are known or constrained — and we use these as compatibility equations to solve the redundant forces.

For example, if a beam has an extra support, we assume a redundant force (e.g., a reaction) and compute the resulting deflection at that point using superposition or virtual work. This deflection is then set equal to the required geometric constraint (usually zero), forming the compatibility equation.

The analysis then becomes a two-part process:

  1. Assume a redundant and analyze the structure using standard deflection techniques.
  2. Apply compatibility (e.g., total deflection at the redundant point must be zero) to solve for the redundant force.

Once the redundant force is known, you can fully determine internal forces, reactions, and deflected shapes using the usual methods.

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Indeterminate Beams by Three Moment Equation

Problem: Propped Beam with Trapezoidal Load

Calculate all support reactions of the given beam.

Indeterminate Beams: Assorted Methods | Structural Theory – Problem: Propped Beam with Trapezoidal Load – Diagram Indeterminate Beams: Assorted Methods | Structural Theory – Problem: Propped Beam with Trapezoidal Load – Diagram Indeterminate Beams: Assorted Methods | Structural Theory – Problem: Propped Beam with Trapezoidal Load – Diagram

See images:

Indeterminate Beams: Assorted Methods | Structural Theory – Problem: Propped Beam with Trapezoidal Load – Diagram Indeterminate Beams: Assorted Methods | Structural Theory – Problem: Propped Beam with Trapezoidal Load – Diagram Indeterminate Beams: Assorted Methods | Structural Theory – Problem: Propped Beam with Trapezoidal Load – Diagram Indeterminate Beams: Assorted Methods | Structural Theory – Problem: Propped Beam with Trapezoidal Load – Diagram

Problem: Fixed-Ended Beam with Triangular Load

Solve all of the support reactions of the fixed-ended beam shown with a triangular load of magnitude 20kN/m acting over the whole span.

Indeterminate Beams: Assorted Methods | Structural Theory – Problem: Fixed-Ended Beam with Triangular Load – Diagram Indeterminate Beams: Assorted Methods | Structural Theory – Problem: Fixed-Ended Beam with Triangular Load – Diagram Indeterminate Beams: Assorted Methods | Structural Theory – Problem: Fixed-Ended Beam with Triangular Load – Diagram

See images:

Indeterminate Beams: Assorted Methods | Structural Theory – Problem: Fixed-Ended Beam with Triangular Load – Diagram Indeterminate Beams: Assorted Methods | Structural Theory – Problem: Fixed-Ended Beam with Triangular Load – Diagram Indeterminate Beams: Assorted Methods | Structural Theory – Problem: Fixed-Ended Beam with Triangular Load – Diagram Indeterminate Beams: Assorted Methods | Structural Theory – Problem: Fixed-Ended Beam with Triangular Load – Diagram

Problem: Fixed-Ended Beam with Support at the Middle

Calculate all of the support reactions of the given beam.

Indeterminate Beams: Assorted Methods | Structural Theory – Problem: Fixed-Ended Beam with Support at the Middle – Diagram Indeterminate Beams: Assorted Methods | Structural Theory – Problem: Fixed-Ended Beam with Support at the Middle – Diagram Indeterminate Beams: Assorted Methods | Structural Theory – Problem: Fixed-Ended Beam with Support at the Middle – Diagram

See images:

Indeterminate Beams: Assorted Methods | Structural Theory – Problem: Fixed-Ended Beam with Support at the Middle – Diagram Indeterminate Beams: Assorted Methods | Structural Theory – Problem: Fixed-Ended Beam with Support at the Middle – Diagram Indeterminate Beams: Assorted Methods | Structural Theory – Problem: Fixed-Ended Beam with Support at the Middle – Diagram Indeterminate Beams: Assorted Methods | Structural Theory – Problem: Fixed-Ended Beam with Support at the Middle – Diagram Indeterminate Beams: Assorted Methods | Structural Theory – Problem: Fixed-Ended Beam with Support at the Middle – Diagram Indeterminate Beams: Assorted Methods | Structural Theory – Problem: Fixed-Ended Beam with Support at the Middle – Diagram Indeterminate Beams: Assorted Methods | Structural Theory – Problem: Fixed-Ended Beam with Support at the Middle – Diagram

Indeterminate Beams by Superposition Method

Problem: Propped Beam with Parabolic Load

The beam shown is subjected to a parabolic distributed load. Determine the location and magnitude of the maximum deflection. EI = constant

Indeterminate Beams: Assorted Methods | Structural Theory – Problem: Propped Beam with Parabolic Load – Diagram

Steps in solving the reactions by the superposition method:
1. Remove the roller support at B
2. Solve the maximum deflection if the roller is removed
3. Solve the equivalent load at B so that the deflection at the free end is completely counteracted, assuming there is no support settlement (compatibility equation). This will be the reaction at B.

Indeterminate Beams: Assorted Methods | Structural Theory – Problem: Propped Beam with Parabolic Load – Diagram Indeterminate Beams: Assorted Methods | Structural Theory – Problem: Propped Beam with Parabolic Load – Diagram Indeterminate Beams: Assorted Methods | Structural Theory – Problem: Propped Beam with Parabolic Load – Diagram Indeterminate Beams: Assorted Methods | Structural Theory – Problem: Propped Beam with Parabolic Load – Diagram Indeterminate Beams: Assorted Methods | Structural Theory – Problem: Propped Beam with Parabolic Load – Diagram
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