Elastic beam: four analytic cases

From the free body to the equation

Begin with the support model. A pin and a roller prevent vertical translation but allow rotation; the reactions share a symmetric load equally. A left clamp prevents both translation and rotation, and supplies a force and a counterclockwise reaction couple. Its external couple is opposite to the internal end bending moment.

Cut the beam at x and balance the part to the left. This gives shear V and sagging-positive moment M. Here dM/dx = V and dV/dx = −w between concentrated forces. A point force creates a jump in V; M, slope and displacement remain continuous when no concentrated couple is applied.

Euler–Bernoulli kinematics keeps plane cross-sections normal to the deflected centreline. With small slopes, curvature is d²v/dx². Linear elasticity gives EI d²v/dx² = M. Integrating twice and enforcing support conditions gives the four expressions below; no mesh or iterative solver is involved.

The coordinate x starts at the left support or clamp. Positive applied P or w is downward; positive v is upward. A downward load therefore gives a negative displacement. Positive M is sagging. The shear sign is defined by dM/dx = V. The diagrams use separate vertical scales and magnify deformation; they are not a drawing of physical displacement to scale.

Slenderness cannot be established from I and L alone. You must know the actual cross-section depth, the bending axis and the load path. A deep beam, soft shear material, flexible support or large displacement needs a different model. This tool reports |v|max/L and maximum |θ| without guessing a universal limit.

For a shipboard teaching exercise, a short equipment-support beam can be idealised only after checking how it is attached. Distributed dead load may be represented by w, including self-weight if you supply it. The example omits vibration, impact, wave accelerations and local bracket details; it is not a load case for an actual vessel.

Worked maritime-context example

Synthetic training case: an ideal simply supported equipment-support beam has L = 4 m, E = 200 GPa, I = 8000 cm⁴ and a 20 kN downward point force at midspan. These numbers do not describe a verified installed component. The example button loads the exact numbers for the selected case; all four case examples use the same declared stiffness.

Calculated response

Simply supported · centre point load

Displayed results are rounded. Validation and CSV retain the full converted floating-point precision.

Formal linear-model response. These ratios describe the calculated response. They are not pass/fail thresholds.

Active analytic equations

• RA=RB=P/2; MA=0
• V=P/2 (x<L/2); V=-P/2 (x>L/2)
• a=min(x,L-x); M=P a/2
• v=-P a(3L^2-4a^2)/(48EI)
• theta=-P(3L^2-12x^2)/(48EI) for x<=L/2; theta(x)=-theta(L-x) for x>L/2
• curvature=M/(EI)
• |v|max=|P|L^3/(48EI); x=L/2

Substituted calculation

1. L = 4 × 1 = 4 m
2. E = 200 × 1.0000000e+9 = 2.0000000e+11 Pa; I = 8,000 × 1.0000000e-8 = 0.00008 m⁴
3. P = 20 × 1,000 = 20,000 N; EI = 2.0000000e+11 × 0.00008 = 16,000,000 N·m²
4. W = P = 20,000 N
5. RA = RB = W/2 = 20,000/2 = 10,000 N; MA = 0
6. M(L/2) = PL/4 = 20,000 N·m
7. v(2) = −(1/48) × (20,000) × 4^3 / 16,000,000 = -0.00166666667 m = -1.66666667 mm
8. |v|max/L = 0.00166666667/4 = 0.000416666667; max |θ| = 0.00125 rad

Every listed expression should be zero, up to floating-point roundoff. These are algebraic model checks, not a physical validation or code check.

Assumptions and boundaries

• Straight, slender, prismatic member; constant E and I; linear elastic response and small displacement/rotation. No cross-section, elastic-limit or slenderness certification is attempted.
• Only the four displayed cases are supported. No partial uniform load, off-centre point load, multiple load superposition, continuous span, spring support or variable stiffness is silently approximated.
• No shear deformation, torsion, axial stress, thermal strain, instability, fatigue or dynamic amplification. Member self-weight is excluded unless included in the entered uniform load.
• Ideal supports are bilateral mathematical constraints. Upward loading may require real hold-downs; a roller that can lift off does not satisfy that assumption.
• Residuals test algebra and support compatibility. A zero residual does not prove that the chosen model represents the real structure.
• Numerical bounds are implementation limits, not operating limits: SI L 10⁻⁶–10⁴ m, E 10³–10¹³ Pa, I 10⁻²⁴–10⁶ m⁴; load is zero or |load| 10⁻¹²–10¹⁵ N (or N/m). Derived |v| and |θ| are limited to 10¹² in SI. These broad ranges do not establish physical validity.

Teaching calculation only. No capacity, safety or regulatory approval is implied.

Check your understanding

Sources and curriculum connection

Curriculum connection: İTÜ GEM203 covers internal forces, shear/moment diagrams and bending; MUK204 covers beam elastic curves. These are bounded teaching applications of those topics, not the full course or an institution-approved design tool.

• MIT OCW · David Roylance · Beam Displacements (2000), pp. 1–3, 6–7, 9
• İTÜ · GEM203 · Strength of Materials I
• İTÜ · MUK204 · Strength of Materials II

Primary source content checked 9 October 2026. Historical mechanics publications support the equations; they are not current design codes.