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Mohr’s moment-area theorems

Step through the two theorems, then drag points A and B along the beam. The tangents, the shaded M/EI area and its centroid move with them, and every number is worked out below.

Step 1 of 5

The idea: curvature comes from bending moment

A beam bends into an elastic curve. Its curvature at any section is d²y/dx² = M/EI. So if we know the M/EI diagram, we know how fast the slope is changing everywhere. Mohr turned this into two simple rules about areas.

Beam

Points A and B

You can also drag the A and B markers on the drawing.

Elastic curve (deflection exaggerated)ABM/EI diagram (1/m) · shaded: area between A and B

Theorem 1: change of slope

  1. Area of the M/EI diagram from A to B∫ M/EI dx from x = 0 to 6 m

    Area = −9.9875 × 10⁻³ rad

  2. So the tangents turn by that muchθB − θA = Area

    θA = 0 rad, θB = −0.01 rad

    θB − θA = −0.01 rad ✓ equals the area (−0.573°)

Theorem 2: tangential deviation

  1. Centroid of the area, measured from Bx̄B = moment of area about Barea

    x̄B = −0.046592−9.9875 × 10⁻³ = 4.665 m

  2. Deviation of B from the tangent at At(B/A) = Area × x̄B

    t(B/A) = −9.9875 × 10⁻³ × 4.665 = −46.5917 mm

    Check on the curve: yB − yA − θA(xB − xA) = −46.6667 − (0) − (0) × 6 m = −46.6667 mm ✓

The two deviations are generally different, because the same area has a different lever arm about A and about B.

Applying it: deflection at B

  1. Theorem used

    At the fixed end the tangent to the elastic curve stays horizontal, so the deflection at any point equals the tangential deviation from that tangent.

    Mohr’s 2nd theorem: deviation = (moment of the M/EI diagram between the two points, taken about the point) .

  2. Area of the BM diagram between x = 0 and 6 m

    A = ∫ M dx = −159.7334 kN·m²

    Its centroid is at x̄ = 1.3356 m, i.e. 4.6644 m from the point

  3. Deflection

    EI·y = A × distance = −159.7334 × 4.6644 = −745.0669 kN·m³

    y = −745.066916000 = −46.567 mm (downward)

Actual deflection at B from the elastic curve: -46.667 mm. For any loading and support, see the full SFD and BMD solver.