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:
Assume a redundant and analyze the structure using standard deflection techniques.
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.
Indeterminate Beams by Three Moment Equation
Problem: Propped Beam with Trapezoidal Load
Calculate all support reactions of the given beam.
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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.
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Problem: Fixed-Ended Beam with Support at the Middle
Calculate all of the support reactions of the given beam.
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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
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 by Integration (Point Load Analogous)
Problem: Fixed-Ended Beam with Irregular Loading Given the Equation of the Curve
For the beam shown, determine the following:
a. The moment at A in kN-m
b. The moment at B in kN-m
c. The vertical reaction at B in kN
Indeterminate Beams by Double Integration Method
Problem: Fixed-Ended Beam with Internal Hinge
For the beam shown, determine all of the support reactions and draw the shear and moment diagram.
