Example 1

The rectangular cross-sectioned cantilever beam is loaded at the end with a force P = 10.0 kN inclined at an angle α = 30° to the vertical axis (see the drawing next to it). Determine the distribution of normal stresses in the support section and the position of the neutral axis. Design the beam's cross-section if the allowable stresses are σ = 150 MPa.

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Solution

Classic Version

Force distribution P into components

𝑃𝑦=𝑃⁢𝑠⁢𝑖⁢𝑛⁢𝛼=5⁢𝑘⁢𝑁𝑃𝑦=𝑃⁢𝑐⁢𝑜⁢𝑠⁢𝛼=8,66⁢𝑘⁢𝑁

Load in the xz and xy planes, bending moment diagrams

𝑀𝑦=−𝑃𝑧⋅6−12⋅5⋅62=−141,96⁢𝑘⁢𝑁⁢𝑚 𝑀𝑧=−𝑃𝑦⋅6=−26 𝑘⁢𝑁⁢𝑚

Moment vectors, identification of characteristic section points

Normal stresses

𝜎=𝑀𝑦𝐼𝑦⁢𝑧+𝑀𝑧𝐼𝑧⁢𝑦𝐼𝑦=𝑏⁢ℎ312=2⁢𝑎⋅(3⁢𝑎)312=92⁢𝑎4𝐼𝑧=ℎ⁡𝑏312=3⁢𝑎⋅(2⁢𝑎)312=2⁢𝑎4

Determining the neutral axis equation

𝜎=0𝜎𝑚⁢𝑎⁢𝑥=141,92⋅10392⁢𝑎4⋅1,5⁢𝑎+30⋅1032⁢𝑎4⋅𝑎𝑧=−0,475⁢𝑦

Two points are enough to plot a linear function

𝑦=0,𝑧=0⁢𝑦=1,𝑧=−0,475

Strength condition σ_max≤k_g

𝜎𝑚⁢𝑎⁢𝑥=141,92⋅10392⁢𝑎4⋅1,5⁢𝑎+30⋅1032⁢𝑎4⋅𝑎≤𝑘𝑔𝑎=0,075⁢𝑚=7,5⁢𝑐⁢𝑚

Stresses at characteristic points

\begin{aligned} &σ_A=147,72MPa\ &σ_B=-76,61MPa\ &σ_C=-147,72MPa\ &σ_D=76,61MPa\ \end{aligned

Normal stress diagram