What does this calculator solve?

A prismatic fixed–fixed beam subject to a single concentrated bending moment M at distance a from the left end. It reports reactions, end moments, internal moment, slope, deflection, shear, and stress.

Is shear force zero for a pure couple?

Global net shear is zero, but to satisfy the fixed-end restraints, equal and opposite end shears occur; the internal moment diagram jumps by M at x=a and varies linearly elsewhere.

Which sign convention is used?

Moments are positive when compressing the top fibers. Deflection and shear positive upward. Slope positive up and to the right.

AMESWEB

Fixed–Fixed Beam — Single Moment Calculator

Solve a fixed–fixed beam with a single concentrated bending moment M applied at position a from the left end.

Fixed–fixed beam with a single applied bending moment at position a from the left end

Global units

Length

Moment

Force

Stress

Slope

E (modulus)

I (inertia)

Deflection length (y)

These selections apply to all inputs and results unless noted. Deflection (y) and fiber distance c have their own unit pickers.

Input parameters

Applied moment M_o Unit: lbf·in

Beam length L Unit: ft

Moment position a (from left end) Unit: ft

Report position x Unit: ft

Elastic modulus E Unit: ksi

Distance to extreme fiber c

Unit for c:

Second moment of area I Unit: in⁴

Use dot “.” as decimal separator.
Sign convention: moments positive when compressing top fibers; shear/deflection positive upward; slope positive up & right.

Numbers and units update together.

Need I for common shapes? Try the sectional properties calculators.

Results

Parameter	Value
Reaction Force R₁	--- lbf
Reaction Force R₂	--- lbf
Shear @ x (Vₓ)	--- lbf
Max Shear (Vmax)	--- lbf
Reaction Moment Left (M₁)	--- lbf·in
Reaction Moment Right (M₂)	--- lbf·in
Moment @ x (Mₓ)	--- lbf·in
Max Moment (Mmax)	--- lbf·in
Slope @ x (θₓ)	--- radian
Max Slope (θmax)	--- radian
End Slope Left (θ₁)	--- radian
End Slope Right (θ₂)	--- radian
Deflection @ x (yₓ)	--- ft
Max Deflection (ymax)	--- ft
End Deflection Left (y₁)	--- ft
End Deflection Right (y₂)	--- ft
Bending Stress @ x (σₓ)	--- psi
Max Bending Stress (σmax)	--- psi

Charts

Moment, shear, slope, and deflection plots update after calculation.

About this load case

This calculator treats a fixed–fixed beam with a single applied bending moment M at position a from the left end.

Key behavior

Moment diagram: linear in each span with a jump of magnitude M at x=a (sign per convention). End moments develop to satisfy zero rotations.
Shear: equal and opposite end shears (R₁ = -R₂) maintain global force equilibrium.
Stress: σ(x) = M(x) c / I.

Formulas used (Euler–Bernoulli)

x measured from the left end; constant E and I.

End reactions/moments: closed form in terms of M, a, L (as implemented in the code-behind).
Slope: θ(x) = θ(0) + ∫ M/(E I) dx, piecewise (fixed ends enforce θ(0)=θ(L)=0).
Deflection: y(x) = ∫ θ(x) dx with y(0)=y(L)=0.
Shear: V(x) = dM/dx piecewise; jump in moment at a.

Assumptions & limits

Linear elastic, small deflection, prismatic beam; constant E and I.
Single concentrated moment only; no axial or distributed loads.
Sign convention as stated above.

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