Pin-Roller Supports
Beam supported by a pin at one end and roller at the other. Most common structural configuration. Allows rotation at both ends and horizontal movement at one end.
| Loading Case | Diagram | Max Shear (V) | Max Moment (M) | Max Deflection (δ) | Location |
|---|---|---|---|---|---|
| Central Point Load | P at L/2 | Vmax = P/2 | Mmax = PL/4 | δmax = PL3/(48EI) | δ at x = L/2 |
| Point Load (any position) | P at a | Vmax = Pb/L (at A) | Mmax = Pab/L (at load) | δmax = Pb(L2-b2)3/2/(9√3EIL) | δ at x = √(L2-b2)/3 |
| Uniformly Distributed | w (N/m) | Vmax = wL/2 (at supports) | Mmax = wL2/8 (at center) | δmax = 5wL4/(384EI) | δ at x = L/2 |
| Triangular Load (symmetric) | w0 at center | Vmax = w0L/4 (at supports) | Mmax = w0L2/12 (at center) | δmax = w0L4/(120EI) | δ at x = L/2 |
| Moment at one end | M0 at A | V = M0/L (constant) | Mmax = M0 (at A) | δmax = M0L2/(9√3EI) | δ at x = L/√3 |
Fixed-Free End
Beam fixed at one end and free at the other. Common in balconies, canopies, and brackets. Maximum moment and deflection occur at different locations than simply supported beams.
| Loading Case | Diagram | Max Shear (V) | Max Moment (M) | Max Deflection (δ) | Location |
|---|---|---|---|---|---|
| Point Load (at free end) | P at B | Vmax = P (constant) | Mmax = PL (at fixed) | δmax = PL3/(3EI) | δ at x = L (free end) |
| Point Load (any position) | P at a | V = P (0 to a) | Mmax = Pa (at fixed) | δmax = Pa2(3L-a)/(6EI) | δ at x = L (free end) |
| Uniformly Distributed | w (N/m) | Vmax = wL (at fixed) | Mmax = wL2/2 (at fixed) | δmax = wL4/(8EI) | δ at x = L (free end) |
| Triangular Load (max at fixed) | w0 at A | Vmax = w0L/2 (at fixed) | Mmax = w0L2/6 (at fixed) | δmax = w0L4/(30EI) | δ at x = L (free end) |
| Moment at free end | M0 at B | V = 0 | M = M0 (constant) | δmax = M0L2/(2EI) | δ at x = L (free end) |
Both Ends Fixed
Beam with both ends rigidly fixed against rotation. Stiffer than simply supported with lower deflections. Indeterminate structure requiring compatibility equations.
| Loading Case | Diagram | Max Shear (V) | Max Moment (M) | Max Deflection (δ) | Location |
|---|---|---|---|---|---|
| Central Point Load | P at L/2 | Vmax = P/2 (at supports) | Mmax = PL/8 (at supports & center) | δmax = PL3/(192EI) | δ at x = L/2 |
| Uniformly Distributed | w (N/m) | Vmax = wL/2 (at supports) | Mmax = wL2/12 (at supports) | δmax = wL4/(384EI) | δ at x = L/2 |
| Point Load (any position) | P at a | VA = Pb2(3a+b)/L3 | MA = Pa2b/L2, MB = Pab2/L2 | δ at load = Pa3b3/(3EIL3) | δ at x = a |
Fixed-Pin Supports
Beam fixed at one end and simply supported at the other. Statically indeterminate (degree 1). Combines characteristics of cantilever and simply supported beams.
| Loading Case | Diagram | Max Shear (V) | Max Moment (M) | Max Deflection (δ) | Location |
|---|---|---|---|---|---|
| Point Load (center) | P at L/2 | Vmax = 11P/16 (at pin) | Mmax = 3PL/16 (at fixed) | δmax = 0.0093PL3/EI | δ at x ≈ 0.45L |
| Uniformly Distributed | w (N/m) | Vmax = 5wL/8 (at pin) | Mmax = wL2/8 (at fixed) | δmax = wL4/(185EI) | δ at x ≈ 0.42L |