Beam Configuration
Simply Supported Beam
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.
Load Cases
5 cases
Determinacy
Statically Determinate
Loading Cases & Formulas
Linear Elastic| 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 |
Variable Definitions
P
Point load (N)
w, w0
Distributed load (N/m)
L
Beam length (m)
E
Young's modulus (Pa)
I
Moment of inertia (m4)
EI
Flexural rigidity
Beam Configuration
Cantilever Beam
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.
Load Cases
5 cases
Determinacy
Statically Determinate
Loading Cases & Formulas
Linear Elastic| 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) |
Variable Definitions
P
Point load (N)
w, w0
Distributed load (N/m)
L
Beam length (m)
E
Young's modulus (Pa)
I
Moment of inertia (m4)
EI
Flexural rigidity
Beam Configuration
Fixed-Fixed Beam
Both Ends Fixed
Beam with both ends rigidly fixed against rotation. Stiffer than simply supported with lower deflections. Indeterminate structure requiring compatibility equations.
Load Cases
3 cases
Determinacy
Indeterminate (3rd degree)
Loading Cases & Formulas
Linear Elastic| 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 |
Variable Definitions
P
Point load (N)
w, w0
Distributed load (N/m)
L
Beam length (m)
E
Young's modulus (Pa)
I
Moment of inertia (m4)
EI
Flexural rigidity
Beam Configuration
Propped Cantilever
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.
Load Cases
2 cases
Determinacy
Indeterminate (1st degree)
Loading Cases & Formulas
Linear Elastic| 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 |
Variable Definitions
P
Point load (N)
w, w0
Distributed load (N/m)
L
Beam length (m)
E
Young's modulus (Pa)
I
Moment of inertia (m4)
EI
Flexural rigidity