Timoshenko Beam Simulator — Tip Deflection with Shear Deformation
Compare cantilever tip deflection from Euler-Bernoulli and Timoshenko theories. Vary length, depth, load and Young's modulus to see why short, deep beams need shear deformation.
Blue = Euler-Bernoulli deflected shape / Red = Timoshenko deflected shape (exaggerated) / left: clamped end hatching, right: tip load P
Theory & Key Formulas
For a cantilever of length $L$ with a tip point load $P$, the tip deflection is the sum of a bending contribution and a shear contribution.
Euler-Bernoulli tip deflection (bending only):
$$\delta_\text{EB} = \frac{P L^3}{3 E I}$$
Timoshenko tip deflection (bending + shear):
$$\delta_\text{T} = \frac{P L^3}{3 E I} + \frac{P L}{\kappa G A}$$
Shear modulus $G$ (Poisson's ratio $\nu$):
$$G = \frac{E}{2(1+\nu)}$$
For a rectangular section $b \times h$, $A = bh$ and $I = bh^3/12$. The shear correction factor for a rectangle is $\kappa = 5/6$. The smaller the slenderness ratio $L/h$, the larger the shear contribution ratio $\delta_\text{shear}/\delta_\text{T}$.
What is the Timoshenko Beam Simulator
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In our textbook, the tip deflection of a cantilever is given by $\delta = PL^3/(3EI)$. Is there really a different theory called Timoshenko?
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That formula is from Euler-Bernoulli theory, which only accounts for bending. In reality, when a beam deforms, the cross sections also shear sideways a little. Timoshenko theory is the more refined version that includes that shear deformation too. In the simulator, look at the blue (EB) and red (Timoshenko) curves side by side — the red one sits just slightly below the blue.
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But with the default settings, the shear contribution is only 0.77%. Is it really OK to ignore it?
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Good question. The default slenderness ratio L/h = 10 is for a fairly slender beam. Try setting the length to 0.20 m and the depth to about 200 mm. Once L/h drops below 1, you have a "stubby" beam, and the shear contribution jumps dramatically. Short, deep beams are dominated by shear.
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Wow, you're right! Once L/h drops below 3, the contribution goes above 10%. How do engineers decide which theory to use in practice?
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A common rule is to use Timoshenko when L/h is less than 10. Things like automotive suspension arms, stiffeners on bridge girders, composite sandwich panels and local members of I-beams need it. In FEM, elements like ANSYS BEAM188 or Abaqus B31 are Timoshenko-based by default — it's the standard choice.
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I see. There's even a yellow "Short stocky beam: use Timoshenko" warning. What is this shear correction factor κ?
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κ corrects for the fact that the shear stress is not uniform across the section. The true distribution is parabolic — maximum at the neutral axis, zero at the top and bottom — but Timoshenko theory uses an average, so we add this correction factor. It's 5/6 for a rectangle, 9/10 for a circle, around 0.4 for an I-section. This simulator fixes the section to a rectangle and uses κ = 5/6, so just remember "κ depends on the section shape" for now.
Frequently Asked Questions
Euler-Bernoulli theory assumes that cross sections stay perpendicular to the neutral axis after deformation, which effectively makes them infinitely rigid in shear. In reality, sections shear into a parallelogram-like shape and produce an additional displacement. That extra displacement is the shear contribution and adds the term PL/(κGA) to the bending contribution for a cantilever. The shorter the beam, the more dominant shear becomes, so the contribution grows.
As a guideline, below L/h = 10 shear deformation can no longer be ignored, and below L/h = 5 it is the dominant contribution. Above L/h = 20, Euler-Bernoulli theory is practically sufficient. Composite beams with a high E/G ratio require Timoshenko theory even at larger slenderness, and sandwich panels and laminated composites need particular care.
Commercial FEM packages such as ANSYS, Abaqus and Nastran use Timoshenko-based beam elements as the standard. ANSYS BEAM188, Abaqus B31 and Nastran CBEAM are all Timoshenko-type and handle slender to short beams with the same element. The classical Euler-Bernoulli element is used only when the beam is explicitly known to be slender. Timoshenko is the safer general-purpose choice.
Yes, very much. Ignoring shear deformation in Euler-Bernoulli theory overestimates the higher natural frequencies. Even the first mode can have a few percent of error for short beams, and the 2nd and 3rd modes can be overestimated by 10% to 30%. The full Timoshenko beam theory, which also includes rotary inertia, agrees well with experiments. It is the standard choice for vibration analysis of turbine blades, high-rise buildings and similar structures.
Real-World Applications
Sandwich panels and composite structures: Sandwich panels used in aircraft wings and ship decks are thin face sheets bonded to a lightweight core. Because the core has low shear stiffness compared to the face sheets, the shear contribution is significant even when the structure looks slender. Designers use Timoshenko theory or higher-order shear deformation theory to evaluate deflection and natural frequencies.
Bridge and girder design: In I-section steel and box girder bridges, the web's shear deformation cannot be neglected even in long spans. Particularly for local stress evaluation near supports or for accurate deflection prediction under live load, models that include shear deformation are used. Bridge codes also include correction terms when the depth-to-span ratio is large.
Vibration analysis of machine elements: Turbine blades, crankshafts and machine tool spindles all use Timoshenko theory in their dynamic analysis. Higher vibration modes need accurate prediction, so formulations that account for both shear deformation and rotary inertia are standard. This directly affects the accuracy of resonance avoidance design.
Seismic response analysis: Reinforced concrete columns and beams show pronounced bending-shear interaction in short-span members. Seismic design uses inelastic beam elements that include shear deformation (e.g., multi-axial spring models) to evaluate story drift and plastic hinge formation. Classical Euler-Bernoulli elements can give unconservative results in this context.
Common Misconceptions and Cautions
The most common misconception is to think that Timoshenko theory is a niche topic and Euler-Bernoulli is enough in practice. In reality, the beam elements in commercial FEM packages (BEAM188, B31, CBEAM, etc.) are Timoshenko-based by default and represent the mainstream of modern beam analysis. Euler-Bernoulli is a special case limited to slender beams and gives wrong results for short beams, composites and vibration analysis. The general-purpose choice is Timoshenko. Watching the shear contribution change as you vary the slenderness in this simulator should help you re-frame Timoshenko as "the more general theory", not as a niche method.
The next common mistake is to treat the shear correction factor κ as always 5/6. κ depends strongly on the cross-section shape: about 5/6 ≈ 0.833 for a rectangle, 9/10 = 0.9 for a circle, around 0.5 for a thin-walled tube, and 0.3 to 0.5 for an I-section depending on the web dimensions. This simulator fixes the section to a rectangle, but if the real member is an I-beam or pipe, using the wrong κ will over- or underestimate the shear contribution. Formulas for κ for various sections (e.g., Cowper 1966) are tabulated in the literature, so always consult them when designing.
Finally, do not oversimplify the assumption that the bending and shear contributions can simply be added. This simulator uses ideal conditions — linear, small-deformation, isotropic material, rectangular section — to add the two contributions. In real structures, second-order coupling between bending moment and shear, section warping, material nonlinearity and buckling all interact. In particular, short RC columns and steel plate shear failures are governed by bending-shear interaction at the failure stage. Treat the results here as the entry point of first-order Timoshenko theory within the elastic, linear range.