Eccentric Load Calculator: Axial Load and Bending Stress at a Section

How this calculator works

Choose a rectangle, or enter the area and second moments of any section. Enter the axial load (tension or compression), how far it acts off the centroid in each direction, and any extra bending moments. The calculator gives the stress at the four corners from N/A plus M y/I plus M x/I.

It also gives the shear stress, and if you enter a stress limit, the largest load that keeps the section within it. It handles brackets, short columns, built-up sections such as a T, and two-way eccentricity such as an angle or a beam loaded off its axis.

It warns when a compressive load puts part of the section into tension, which matters for concrete, masonry and soil, and for the middle-third rule.

Formulas used

• \sigma = \frac{N}{A} + \frac{M_x y}{I_x} + \frac{M_y x}{I_y}
• \tau_{max} = \frac{3V}{2A}\ (\text{rectangle})

Worked example

With the default numbers in the calculator above:

1. Section properties: A = 10800.00 mm²
2. Axial stress: 2.22 MPa
3. Total bending moments about x and y (load eccentricity plus the extra moments): 5.40 and 0.00 kN·m
4. Stress at the top right corner: 18.89 MPa
5. Stress at the top left corner: 18.89 MPa
6. Stress at the bottom right corner: -14.44 MPa
7. Stress at the bottom left corner: -14.44 MPa
8. Greatest tension and greatest compression: 18.89 MPa (tension) and -14.44 MPa (compression)
9. Maximum transverse shear stress (rectangle: one and a half times the average): 2.50 MPa

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: Section Any section (enter its properties); Axial load 50 kN; The axial load is Compression (pushing); Eccentricity of the load in y (positive towards the top) -212.5 mm; Extra bending moment about the x axis (tension at the top if positive) 0 kN·m; Transverse shear force 0 kN; Stress limit to find the allowable load for (0 to skip; the size of the worst stress of its kind) 6 MPa.

1. Section properties: A = 90000.00 mm²
2. Axial stress: -0.56 MPa
3. Total bending moments about x and y (load eccentricity plus the extra moments): 10.63 and 0.00 kN·m
4. Stress at the top right corner: 0.72 MPa
5. Stress at the top left corner: 0.72 MPa
6. Stress at the bottom right corner: -2.34 MPa
7. Stress at the bottom left corner: -2.34 MPa
8. Greatest tension and greatest compression: 0.72 MPa (tension) and -2.34 MPa (compression)
9. Largest load that keeps the stress within 6.00 MPa (compression or tension, whichever governs): 128.08 kN at the bottom right corner

Common questions

Related calculators

• Section Properties Calculator: Area, Second Moment and Section Modulus
• Euler Column Buckling Calculator
• Axial Stress, Strain and Elongation Calculator

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Hibbeler or Gere, Mechanics of Materials (eccentric axial loading).

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