Example 1

To draw the diagrams of bending moments, shear forces, and axial forces for a statically indeterminate structure shown in the figure, using the Maxwell-Mohr method to determine hyperstatic reactions.

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Solution

TASK PROCEDURE - method of forces

SSN=4-3=1

Basic System of the Method of Forces (BSMF)

State 𝑋1 =1

Bending moment function

𝑥1∈(0;𝑙)𝑀1⁡(𝑥1)=−1⋅𝑥1𝑥2∈(0;4⁢𝑙)𝑀1⁡(𝑥2)=−𝑙

State P

Bending moment function

𝑥1∈(0;2⁢𝑙)𝑀𝑝⁡(𝑥1)=0𝑥2∈(0;4⁢𝑙)𝑀𝑝⁡(𝑥2)=−𝑃⋅𝑥2

Canonical equation of the Method of Forces

𝛿11⋅𝑥1+𝛿1⁢𝑝=0𝐸⁢𝐼⁢𝛿11=∫𝐿0(𝑀1⋅𝑀1)𝑑𝑥=∫𝑙0𝑥2𝑑𝑥+∫4⁢𝑙0𝑙2𝑑𝑥=13⁢𝑙3+4⁢𝑙3=133⁢𝑙3𝛿11=133⁢𝐸⁢𝐼⁢𝑙3𝐸⁢𝐼⁢𝛿1⁢𝑝=∫𝐿0(𝑀1⋅𝑀𝑝)𝑑𝑥=∫4⁢𝑙0𝑃⁢𝑙⁢𝑥𝑑𝑥=𝑃⁢𝑙⋅12⋅(4⁢𝑙)2=8⁢𝑃⁢𝑙3𝛿1⁢𝑝=8⁢𝑃⁢𝑙3𝐸⁢𝐼𝑥1=−8⁢𝑃⁢𝑙3133⁢𝑙3=−2413⁢𝑃

After calculating the statically indeterminate reactions, we calculate and draw the final diagrams.

𝑥1∈(0;𝑙)𝑁⁡(𝑥1)=𝑃𝑇⁡(𝑥1)=0𝑀⁡(𝑥1)=0𝑥2∈(𝑙;2⁢𝑙)𝑁⁡(𝑥2)=𝑃𝑇⁡(𝑥2)=2413⁢𝑃𝑀⁡(𝑥2)=2413⁢𝑃⁢(𝑥2−𝑙)𝑀⁡(𝑙)=0𝑥3∈(0;4⁢𝑙)𝑁⁡(𝑥3)=2413⁢𝑃𝑇⁡(𝑥3)=−𝑃𝑀⁡(𝑥3)=−2813⁢𝑃⁢𝑙+𝑃⁢𝑥𝑀⁡(0)=−2813⁢𝑃⁢𝑙𝑀⁡(2⁢𝑙)=2413⁢𝑃⁢𝑙

Final internal forces diagrams