Engineering
Simply supported beam deflection (5wL^4/384EI), bending moment & L/360 check
Explore ideal simply supported beam deflection and bending moment, with an illustrative span/360 comparison rather than a structural approval.
Keep SI units consistent in N and mm so beam deflection never misses by 1,000x!
These Euler–Bernoulli beam examples assume a straight, prismatic, simply supported member with constant elastic stiffness and small deflections. They omit shear deformation, buckling, connection behavior, strength and load combinations. They are educational calculations, not a structural design approval.
Inputs use span in m, distributed load in kN/m, elastic modulus in GPa and second moment of area in cm^4. Convert span by 1000, modulus by 1000 and inertia by 10000 to work in N and mm; 1 kN/m equals 1 N/mm. These explicit conversions help checking but cannot prevent incorrect source units. The span/360 comparator is illustrative; applicable limits depend on the element, loading and governing requirements.
| Structural Check (L=6 m, w=12 kN/m, E=200 GPa, I=8,100 cm^4) | Formula (in N & mm) | Result |
|---|---|---|
| Uniform Load Deflection (mm) | 5 * 12 * 6000^4 / (384 * 200000 * 8.1e7) | 12.50 mm |
| Max Bending Moment M_max (kN·m) | w * L^2 / 8 = 12 * 6^2 / 8 | 54.0 kN·m |
| Illustrative span/360 comparison | 6000 mm / 360 = 16.67 mm | Within illustrative limit (12.50 <= 16.67) |
| Center Point Load P = 36 kN Deflection | (36000 * 6000^3) / (48 * E * I) | 10.00 mm |
Midspan deflection (mm) under uniform distributed load w
A2=Span_m (6), B2=Load_kN_m (12), C2=E_GPa (200), D2=I_cm4 (8100).
=IF(OR(COUNT(A2:D2)<>4,A2<=0,B2<0,C2<=0,D2<=0),"Invalid inputs",LET(L, A2 * 1000, w, B2, E, C2 * 1000, I, D2 * 10000, ROUND((5 * w * (L ^ 4)) / (384 * E * I), 2)))Compare uniform-load deflection with an illustrative span/360 limit
Compare unrounded deflection against span/360. The label only describes this numerical comparison; it does not establish safety or code compliance. Uniform load and central point load are separate examples, not combined here.
=IF(OR(COUNT(A2:D2)<>4,A2<=0,B2<0,C2<=0,D2<=0),"Invalid inputs",LET(L, A2 * 1000, EI, (C2 * 1000) * (D2 * 10000), d, (5 * B2 * (L ^ 4)) / (384 * EI), IF(d <= L / 360, "Within illustrative limit", "Exceeds illustrative limit")))Separate central point-load deflection
E2 is the central point load in kN. This assumes the same ideal beam with no distributed load.
=IF(OR(COUNT(A2:D2)<>4,A2<=0,B2<0,C2<=0,D2<=0,NOT(ISNUMBER(E2)),E2<0),"Invalid inputs",LET(L, A2 * 1000, P, E2 * 1000, EI, (C2 * 1000) * (D2 * 10000), ROUND((P * (L ^ 3)) / (48 * EI), 2)))Try the example
Use the original values or change them to test your understanding.