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Influence Line

Influence lines show how internal forces or reactions at a specific location vary as a unit load moves across a structure. They are particularly useful in determining the critical positions of live loads.

Two Common Methods:

1. Tabular Method (Table-Based Analytical Influence Lines)

This method is best suited for statically determinate structures. It involves placing a unit load at key points and solving for the response of the desired function (e.g., support reaction, shear, or moment).

Steps:

  1. Select the function you want to analyze: a support reaction, shear at a section, or moment at a section.
  2. Divide the beam into key load positions — usually at supports and points of interest.
  3. Place a unit load (1.0 kN or 1.0 unit) at one position at a time.
  4. Using equilibrium, solve for the value of the function of interest at each load position.
  5. Plot or tabulate the values to create the influence line diagram.

Example Table Format:

Unit Load Position Support Reaction $R_A$ Shear at Section X Moment at Section X
At A1.0......
Midspan.........
At B0.0......
The tabular method is exact and useful for numerical influence line diagrams, especially for statically determinate systems.

2. Müller-Breslau Principle (Qualitative Influence Lines)

The Müller-Breslau Principle provides a fast, visual way to sketch the shape of an influence line, even for indeterminate structures. It states:

“The influence line for a response function (reaction, shear, or moment) has the same shape as the deflected shape of the structure when the corresponding restraint is released and given a unit displacement in the direction of the function.”

Steps:

  1. Identify the function you want to draw an influence line for (e.g., $R_A$, $V_x$, $M_x$).
  2. Modify the structure by releasing the restraint or cut at the location of the function.
  3. Apply a unit displacement in the direction of the desired function:
    • Vertical displacement for vertical reactions and shear
    • Rotation for moments
  4. Sketch the deflected shape — this is the qualitative influence line.
  5. Positive displacement direction = positive influence value. The relative height of the sketch indicates the magnitude.
The Müller-Breslau Principle is particularly useful for indeterminate structures or quick checks of influence line shapes.

After using the Müller-Breslau sketch, you can optionally apply the conjugate beam method or unit load method to compute exact ordinates if needed.

Concept Concept Concept Concept Concept Concept Concept Concept Concept

Problem:

Refer to the image shown:

Influence Line in Beams and Trusses | Structural Theory – Problem 1: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 1: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 1: – Diagram

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Influence Line in Beams and Trusses | Structural Theory – Problem 1: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 1: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 1: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 1: – Diagram

Problem:

Refer to the image shown:

Influence Line in Beams and Trusses | Structural Theory – Problem 2: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 2: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 2: – Diagram

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Influence Line in Beams and Trusses | Structural Theory – Problem 2: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 2: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 2: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 2: – Diagram

Problem:

Refer to the image shown:

Influence Line in Beams and Trusses | Structural Theory – Problem 3: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 3: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 3: – Diagram

See images:

Influence Line in Beams and Trusses | Structural Theory – Problem 3: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 3: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 3: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 3: – Diagram

Problem:

Refer to the image shown:

Influence Line in Beams and Trusses | Structural Theory – Problem 4: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 4: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 4: – Diagram

See images:

Influence Line in Beams and Trusses | Structural Theory – Problem 4: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 4: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 4: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 4: – Diagram

Problem:

Refer to the image shown:

Influence Line in Beams and Trusses | Structural Theory – Problem 5: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 5: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 5: – Diagram

See images:

Influence Line in Beams and Trusses | Structural Theory – Problem 5: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 5: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 5: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 5: – Diagram

Problem:

Refer to the image shown:

Influence Line in Beams and Trusses | Structural Theory – Problem 6: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 6: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 6: – Diagram

See images:

Influence Line in Beams and Trusses | Structural Theory – Problem 6: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 6: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 6: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 6: – Diagram

Problem:

Refer to the image shown:

Influence Line in Beams and Trusses | Structural Theory – Problem 7: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 7: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 7: – Diagram

See images:

Influence Line in Beams and Trusses | Structural Theory – Problem 7: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 7: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 7: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 7: – Diagram

Problem:

Refer to the image shown:

Influence Line in Beams and Trusses | Structural Theory – Problem 8: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 8: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 8: – Diagram

See images:

Influence Line in Beams and Trusses | Structural Theory – Problem 8: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 8: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 8: – Diagram Influence Line in Beams and Trusses | Structural Theory – Problem 8: – Diagram
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Exam Generator Problems

Additional board-style practice items for this topic.

Question Bank: q5

PSAD - Structural Theory / Influence Line / Engr. Janclyde Espinosa (Clidez)

CE Board May 2010
A simply supported beam 10 meters long has an overhang of 2m at the left support. If a highway uniform load of 9.35kN/m and a concentrated load of 116kN passes through the beam, compute the following based on the influence line for maximum shear at midspan.

Determine the length of the beam where the uniform load could produce maximum positive shear at the midspan.

  1. 7m
  2. 2m
  3. 5m
  4. 10m

Determine the length of the beam where the uniform load could produce maximum negative shear at the midspan.

  1. 5m
  2. 2m
  3. 7m
  4. 10m

If the concentrated load will be placed at the end of the overhang, compute the maximum shear at the midspan.

  1. 23.2kN
  2. 58.4kN
  3. 30kN
  4. 27.9kN

Solution pending in psadquestions/q5.json.

Question Bank: q138

PSAD - Structural Theory / Influence Line / Engr. Janclyde Espinosa (Clidez)

Refer ot the compound bridge girder shown. Analyze the beam using the influence line diagram. The components of the moving load are the following:
Concentrated load = 120kN
Uniform live load = 10kN/m

q138

Compute the ordinate at E considering the inluence line of the maximum reaction at F.

  1. 1.20
  2. 0
  3. 1.00
  4. 1.50

Calculate the maximum reaction at F in kN.

  1. 474
  2. 378
  3. 447
  4. 456

Calculate the maximum reaction at B in kN.

  1. 324
  2. 307
  3. 286
  4. 344

Solution pending in psadquestions/q138.json.

Question Bank: q566

PSAD - Structural Theory / Influence Line / Engr. Deguma

For the beam shown, a=25m, b=5m, and c=15m.
Uniformly Distributed Load, wLL=20kN/m
Concentrated Load = 60kN

q566

For maximum reaction at the support at A, solve for the beam length, which should be loaded with the uniformly distributed load?

  1. 25m
  2. 15m
  3. 10m
  4. 5m

For maximum reaction at the support at D, solve for the beam length, which should be loaded with the uniformly distributed load?

  1. 15m
  2. 25m
  3. 5m
  4. 10m

Which of the following most nearly gives the maximum reaction (kN) at the support at B due to the given loads?

  1. 612
  2. 645
  3. 588
  4. 540

Solution pending in psadquestions/q566.json.

Question Bank: q626

PSAD - Structural Theory / Influence Line / Mastermatician

A 12-meter-long beam is simply supported at the right end and at 2 meters from the left end. It is subjected to a highway load consisting of 9.35 kN/m uniformly distributed load and 116 kN concentrated load. Based on the influence line for maximum moment at midspan,

What is the total length in meters of the beam which should be subjected to a uniformly distributed load to obtain the maximum positive moment at midspan?

  1. 10
  2. 12
  3. 2
  4. 5

What is the total length in meters of the beam which should be subjected to the uniformly distributed load to obtain the maximum negative moment at the midspan?

  1. 2
  2. 5
  3. 10
  4. 12

What would be the critical positive moment (kN-m) at midspan due to the highway lane load?

  1. 407
  2. 214
  3. 325
  4. 154

Solution pending in psadquestions/q626.json.

Question Bank: q628

PSAD - Structural Theory / Influence Line / Mastermatician

A 10 m long beam is simply supported at the left end and at 2 m from the right end. It is to be analyzed for uniformly distributed moving load. For maximum reaction at the right support, what is the total beam length (m) which should be subjected to the moving load?

  1. 10m
  2. 8m
  3. 12m
  4. 6m

Solution pending in psadquestions/q628.json.

Question Bank: q629

PSAD - Structural Theory / Influence Line / Mastermatician

The beam in the figure supports a distributed load of 1.5 kN/m and a single concentrated load of 8 kN. The dead load is 2 kN/m.

q629

Determine the maximum positive moment at C, in kN-m.

  1. 13
  2. 23
  3. 12
  4. 18

Determine the maximum positive shear at C, in kN.

  1. 6.50
  2. 3.45
  3. 4.78
  4. 4.34

Solution pending in psadquestions/q629.json.