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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.

Concept Concept Concept Concept Concept Concept Concept Concept Concept

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

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.

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Exam Generator Problems

Additional board-style practice items for this topic.

Question Bank: q12

PSAD - Structural Theory / Indeterminate Beams / Engr. Janclyde Espinosa (Clidez)

CE Board May 2010
A 6m cantilever retaining wall is loaded with an active pressure which varies from zero at the top to 35kN/m at the bottom. Assume EI is constant.

Compute the maximum shear.

  1. 105
  2. 210
  3. 70
  4. 86

Compute the maximum moment.

  1. 210
  2. 315
  3. 240
  4. 320

If the wall is laterally supported at the free end and fixed at the base, determine the moment at the foundation. Use end moment equation wL²/30 at the top and wL²/20 at the base.

  1. 84
  2. 42
  3. 63
  4. 157.5

Solution pending in psadquestions/q12.json.

Question Bank: q128

PSAD - Structural Theory / Indeterminate Beams / Engr. Janclyde Espinosa (Clidez)

A 4m beam is simply supported at each end and carries a uniform load W = 45kN/m throughout its span.
Beam Properties:
E=25GPa
I=337.5x106mm4

Which of the following gives the maximum shear of the beam?

  1. 90
  2. 33.75
  3. 60
  4. 65.39

Which of the following gives the maximum moment of the beam?

  1. 90
  2. 22.5
  3. 40.78
  4. 37.43

To prevent excessive deflection, an additional support is provided at midspan. Which of the following gives the reaction at the additional support?

  1. 112.5
  2. 0
  3. 60
  4. 90

Part 1.

Simply supported beam, $L=4$ m, $w=45$ kN/m.
Max shear at supports: $V = \frac{wL}{2} = \frac{45 \times 4}{2}$
$\boxed{V_{\max} = 90 \text{ kN}}$

Part 2.

Max moment at midspan: $M = \frac{wL^2}{8} = \frac{45 \times 16}{8}$
$\boxed{M_{\max} = 90 \text{ kN·m}}$

Part 3.

Propped at midspan: compatibility (midspan deflection = 0).
$\frac{5wL^4}{384EI} = \frac{RL^3}{48EI} \Rightarrow R = \frac{5wL}{8} = \frac{5 \times 45 \times 4}{8}$
$\boxed{R = 112.5 \text{ kN}}$

Question Bank: q327

PSAD - Structural Theory / Indeterminate Beams / Engr. Janclyde Espinosa (Clidez)

Using any method of deflection,

q327

Calculate the vertical reaction at B in kN

  1. 12.29
  2. 22.87
  3. 13.82
  4. 20.67

Calculate the moment reaction at A in kN-m

  1. 30.28 (counterclockwise)
  2. 30.28 (clockwise)
  3. 20.83 (clockwise)
  4. 20.83 (counterclockwise)

Calculate the vertical reaction at A in kN

  1. 24.71
  2. 23.82
  3. 29.62
  4. 26.94

Solution pending in psadquestions/q327.json.

Question Bank: q328

PSAD - Structural Theory / Indeterminate Beams / Engr. Janclyde Espinosa (Clidez)

Refer to the beam shown:

q328

Determine the reaction at A in kN

  1. 8.18
  2. 7.18
  3. 6.18
  4. 4.18

Determine the reaction at B in kN

  1. 18.90
  2. 16.80
  3. 17.60
  4. 19.50

Determine the reaction at C in kN

  1. 0.074 (downward)
  2. 0.074 (upward)
  3. 1.323 (downward)
  4. 1.323 (upward)

Determine the internal moment at B in kN-m

  1. 7.297 (counterclockwise)
  2. 7.297 (clockwise)
  3. 6.782 (clockwise)
  4. 6.782 (counterclockwise)

Solution pending in psadquestions/q328.json.

Question Bank: q331

PSAD - Structural Theory / Indeterminate Beams / Engr. Janclyde Espinosa (Clidez)

Determine the reactions of the beam shown using the Three Moment Equation. Note that there is an internal hinge at B. Assume EI is constant.

q331

Determine the deflection at B.

  1. 445.46/EI
  2. 464.54/EI
  3. 376.63/EI
  4. 363.73/EI

Determine the moment reaction at A in kN-m.

  1. 64.12 (CCW)
  2. 64.12 (CW)
  3. 66.57 (CW)
  4. 66.57 (CCW)

Determine the moment reaction at E in kN-m.

  1. 66.57 (CW)
  2. 66.57 (CCW)
  3. 64.12 (CCW)
  4. 64.12 (CW)

Determine the vertical reaction at A in kN.

  1. 28.69
  2. 22.31
  3. 29.86
  4. 23.21

Determine the vertical reaction at E in kN.

  1. 22.31
  2. 28.69
  3. 23.21
  4. 29.86

Solution pending in psadquestions/q331.json.

Question Bank: q545

PSAD - Structural Theory / Indeterminate Beams / Engr. Deguma

For the beam shown, if L = 4 m, and w = 16 kN/m:

q545

Determine the reaction at A in kN.

  1. 8
  2. 12
  3. 16
  4. 20

Determine the reaction at B in kN.

  1. 132
  2. 124
  3. 156
  4. 116

Part 1.

Treat the beam as a two-span continuous beam with spans $AB=L=4$ m and $BC=2L=8$ m. With simple end supports, $M_A=M_C=0$. For uniform load on both spans, the three-moment equation gives:
$2M_B(L_1+L_2)=-6\left(\frac{A_1\bar{x}_1}{L_1}+\frac{A_2(L_2-\bar{x}_2)}{L_2}\right)$
For the simple-span moment diagrams, this gives $M_B=-96$ kN-m. For span AB:
$R_A=\frac{wL_1}{2}+\frac{M_B-M_A}{L_1}$
$R_A=\frac{16(4)}{2}+\frac{-96-0}{4}=8\text{ kN}$
$\boxed{R_A=8\text{ kN}}$

Part 2.

From the left span, the reaction at B is:
$R_{B,left}=wL_1-R_A=16(4)-8=56\text{ kN}$
For span BC, using $M_B=-96$ kN-m and $M_C=0$:
$R_C=\frac{wL_2}{2}+\frac{M_B-M_C}{L_2}$
$R_C=\frac{16(8)}{2}+\frac{-96}{8}=52\text{ kN}$
$R_{B,right}=wL_2-R_C=128-52=76\text{ kN}$
$R_B=56+76=132\text{ kN}$
$\boxed{R_B=132\text{ kN}}$

Question Bank: q611

PSAD - Structural Theory / Indeterminate Beams / Mastermatician

Given the following beam:

q611

Determine the degree of indeterminacy.

  1. 3
  2. 1
  3. 2
  4. 4

Determine the moment reaction at the left end.

  1. 34.5kN-m
  2. 13.5kN-m
  3. 32.25kN-m
  4. 18.75kN-m

Determine the vertical reaction at the middle support.

  1. 39.00kN
  2. 32.25kN
  3. 18.75kN
  4. 13.50kN

Determine the maximum positive moment in the beam.

  1. 17.50kN-m
  2. 34.50kN-m
  3. 6.56kN-m
  4. 13.5kN-m

Solution pending in psadquestions/q611.json.

Question Bank: q612

PSAD - Structural Theory / Indeterminate Beams / Mastermatician

Given the following beam:

q612

Determine the degree of indeterminacy.

  1. 1
  2. 0
  3. 3
  4. 2

Determine the moment reaction at the fixed ends.

  1. 30kN-m
  2. 15kN-m
  3. 45kN-m
  4. 20kN-m

Determine the vertical reaction at the middle support.

  1. 60.00kN
  2. 30.00kN
  3. 45.00kN
  4. 36.50kN

Determine the maximum positive moment in the beam.

  1. 15kN-m
  2. 30kN-m
  3. 45N-m
  4. 60kN-m

Determine the maximum deflection of the beam.

  1. 33.75mm
  2. 36.75mm
  3. 31.25mm
  4. 38.25mm

Solution pending in psadquestions/q612.json.

Question Bank: q613

PSAD - Structural Theory / Indeterminate Beams / Mastermatician

Given the following beam:

q613

Determine the reaction at the roller support in kN.

  1. 24.75kN
  2. 20.25kN
  3. 25.50kN
  4. 19.50kN

Determine the moment reaction at the fixed support in kN-m

  1. 31.5
  2. 22.8
  3. 27.6
  4. 32.9

Determine the location of the point of inflection of the beam.

  1. 1.647m
  2. 4.250m
  3. 1.746m
  4. 4.025m

Determine the maximum deflection of the beam.

  1. 59.25mm
  2. 60.75mm
  3. 53.65mm
  4. 62.75mm

Solution pending in psadquestions/q613.json.

Question Bank: q614

PSAD - Structural Theory / Indeterminate Beams / Mastermatician

An 8-m long propped beam is subjected to a uniform load varying from 70 kN/m at 1 m from the fixed end to 50 kN/m at 2 m from the roller support.

Calculate the vertical reaction at the roller support (kN).

  1. 76.62
  2. 112.43
  3. 223.38
  4. 187.57

Calculate the maximum positive moment within the span.

  1. 209.76
  2. 275.65
  3. 341.54
  4. 395.35

Determine the location of the maximum deflection of the beam from the fixed support, in meters.

  1. 4.46m
  2. 3.17m
  3. 3.54m
  4. 4.83m

Solution pending in psadquestions/q614.json.

Question Bank: q618

PSAD - Structural Theory / Indeterminate Beams / Mastermatician

An 8-m high retaining wall is subjected to lateral earth pressure increasing from 34 kPa at the top to 136 kPa at the base. Flexural rigidity EI = 4.5x1014 N-mm2. Analyze per meter length of the wall.

What is the moment at the base of the cantilever retaining wall, in kN-m?

  1. 2176
  2. 3264
  3. 707
  4. 1469

What is the force to be applied at the propped end to limit the deflection to 35 mm, in kN?

  1. 91.37
  2. 69.63
  3. 92.29
  4. 183.60

What is the moment at the base when the wall is propped at the top?

  1. 707.2
  2. 2176.0
  3. 3644.8
  4. 1468.8

Solution pending in psadquestions/q618.json.

Question Bank: q622

PSAD - Structural Theory / Indeterminate Beams / Mastermatician

The steel beam is supported by a steel rod as shown. A load W is applied at the midspan.
Beam Properties:
I = 198x106mm4
E=200GPa
L=3m
Rod Properties:
Diameter = 12mm
E=200GPa
L=2.4m

q622

Due to the load, W, rod BC elongates by 1 mm. Find the force (kN) in rod BC which caused the elongation.

  1. 9.4
  2. 4.4
  3. 6.4
  4. 11.4

Due to the load, W, the force developed in rod BC is 12 kN, what is the value of W (kN)?

  1. 56.33
  2. 20.47
  3. 6.82
  4. 24.00

Due to a load, W = 40 kN, the force developed in rod BC = 10 kN. The diameter of rod BC is 16 mm. Find the moment (kNm) at the fixed end.

  1. 30
  2. 90
  3. 60
  4. 45

Solution pending in psadquestions/q622.json.