Beam Theory — Engineering Reference

1. At a glance

Beam theory is the branch of mechanics that describes how slender prismatic members carry transverse loads by developing internal bending moments, shear forces, and (in real beams) axial forces and torsion. It is the single most-applied piece of engineering mechanics in industry. Floor joists, roof purlins, bridge girders, crane booms, machine-tool ways, robot links, aircraft wing spars, train axles, ship hulls, wind-turbine blades, even printed-circuit-board carriers — all are analysed first as beams, then refined.

The discipline has two classical formulations:

• Euler–Bernoulli (E–B) beam theory (Euler 1750, Bernoulli ~1700). Plane sections remain plane and perpendicular to the deformed neutral axis; shear deformation is neglected. Excellent for slender beams (span/depth L/h ≳ 10). The basis of every code-book deflection formula and every “beam element” in classical FEA.
• Timoshenko–Ehrenfest beam theory (Timoshenko 1921, refined later). Plane sections remain plane but rotate independently of the neutral axis, capturing transverse shear deformation. Required for deep beams, sandwich panels, short cantilevers, and the high-frequency end of beam-vibration analysis.

Where it sits in the design stack:

• Builds on statics-fundamentals (reactions) and mechanics-of-materials (stress, strain, Hooke’s law).
• Precedes structural analysis of indeterminate beams (force/displacement methods, moment distribution, stiffness matrix), steel and concrete design codes, and the formulation of beam elements in FEA (Bernoulli beam → 2-node Hermite cubic; Timoshenko beam → linear or quadratic Lagrange + shear-locking remedies).

In day-to-day engineering practice: one of strength (σ vs σ_allow) and serviceability (deflection vs L/360 etc.) almost always governs a beam design — only rarely both at once. Knowing which to size against, and when, is the engineering decision the math supports.

2. First principles

Kinematic assumptions — Euler–Bernoulli

E–B beam theory rests on three assumptions, applied to a beam initially straight, with cross-section symmetric about the bending plane:

1. Plane sections remain plane after deformation.
2. Plane sections remain perpendicular to the deformed neutral axis (i.e. transverse shear strain γ = 0).
3. Small deflections — slopes dy/dx ≪ 1, so curvature κ ≈ d²y/dx².

Consequence: every point in the cross-section moves only horizontally (axial direction) due to bending, by an amount proportional to its distance y from the neutral axis (NA). Below the NA the beam stretches; above it shortens (for a sagging beam loaded downward, with +y up).

Bending strain and curvature

For an arc element of beam with radius of curvature ρ, the axial strain at fibre distance y from the NA is:

ε(y) = − y / ρ                  (E–B kinematic relation)

with curvature κ = 1/ρ. For small deflections, κ ≈ d²y/dx² where y(x) is the deflection (transverse displacement) of the neutral axis.

Flexure formula (stress in pure bending)

Applying Hooke’s law σ = E·ε:

σ(y) = − E · y / ρ = − E · y · d²y/dx²

Moment equilibrium about the NA of an infinitesimal cross-section:

M = − ∫ σ · y dA = (E/ρ) · ∫ y² dA = E · I / ρ

where I is the second moment of area (area moment of inertia) about the NA. Combining:

σ(y) = − M · y / I                          (flexure formula)
σ_max = M · c / I = M / S                  (at the extreme fibre)

with:

• c = distance from NA to extreme fibre [m]
• I = second moment of area [m⁴]
• S = I / c = elastic section modulus [m³], the single most-used “shape number” for beams.

The NA passes through the centroid of the cross-section for a homogeneous, linearly-elastic beam in pure bending (no axial force). For unsymmetric sections, use I_z and I_y and the centroidal principal axes.

Shear stress in a beam (Jourawski)

For a transverse shear force V acting on a cross-section, the longitudinal shear stress at a horizontal level distance y₁ from the NA is given by Jourawski’s formula (1856):

τ = V · Q / (I · b)

where:

• Q = first moment of area of the portion of the cross-section above (or below) the level y₁, taken about the NA: Q = ∫_{A’} y dA = ȳ’ · A’
• b = width of the cross-section at level y₁
• I, V as above

For a rectangular section b × h, the shear distribution is parabolic with τ_max = 1.5 · V/A at the NA. For a thin-walled I-beam, the flanges carry most of the bending stress while the web carries virtually all the shear — τ_web,avg ≈ V / A_web is the standard approximation in steel design.

Governing differential equation

Combining flexure with equilibrium dM/dx = V, dV/dx = −w(x) gives the fourth-order beam equation:

EI · d⁴y/dx⁴ = w(x)                  (Euler–Bernoulli, transverse load w(x) per unit length)

Equivalent forms:

EI · d²y/dx² = M(x)                  (when M(x) already known from statics)
EI · d³y/dx³ = V(x)
EI · d⁴y/dx⁴ = − dV/dx = w(x)

Boundary conditions come from the supports: pinned (y = 0, M = 0), fixed (y = 0, dy/dx = 0), free (V = 0, M = 0), roller (y = 0). Solving the ODE with appropriate BCs gives the deflection y(x), from which slope, moment, and shear all follow by differentiation.

Timoshenko correction

E–B drops the term γ = V / (k·G·A), where:

• k = shear-correction factor (form factor for shear); 5/6 for solid rectangle, 0.9 for I-beam web-dominated section, 0.5 for thin circular tube, etc.
• G = shear modulus
• A = cross-sectional area
• V = shear force

Timoshenko beam theory adds an independent rotation ψ(x) of the cross-section, separate from the slope dy/dx of the neutral axis, with γ = dy/dx − ψ. The resulting two coupled ODEs:

d/dx [ EI · dψ/dx ] + k·G·A · (dy/dx − ψ) = 0
d/dx [ k·G·A · (dy/dx − ψ) ] + w(x) = 0

For static loading on a slender beam (L/h > 10), the Timoshenko and E–B deflections differ by < 1–2 %. For deep beams (L/h ≈ 5), the difference can exceed 20 %. Timoshenko also predicts higher natural frequencies more accurately for the second and third bending modes.

Pure bending vs combined loading

The flexure formula assumes pure bending — only M on the section, no N (axial) or V. Real beams almost always carry a combined load. Superposition is valid in the linear-elastic regime:

σ_total(y) = N/A + M·y/I                    (axial + bending)
τ_total = T·r/J + V·Q/(I·b)                  (torsion + transverse shear)

When combined stress matters (column-beams, frame members), use mechanics-of-materials interaction equations (Tresca or von Mises). AISC 360-22 Chapter H prescribes the code interaction equations for combined bending and axial force.

3. Practical math / design equations

3.1 Section modulus and the design workflow

For a beam-sizing problem, the required elastic section modulus is the single most-used design number:

S_req = M_max / σ_allow

Then pick the lightest standard shape from the AISC Manual (US) or the EN profile tables (EU) with S ≥ S_req. For plastic design (LRFD / EC3 ULS), use plastic modulus Z and code-specified factored loads instead.

3.2 Shear-and-moment diagrams

V(x) and M(x) are obtained by integrating the load function:

V(x) = V(0) − ∫₀ˣ w(ξ) dξ           (with appropriate sign for downward w)
M(x) = M(0) + ∫₀ˣ V(ξ) dξ

Key construction rules (the ones every engineer memorizes):

• A point load produces a jump in V equal to its magnitude (downward jump for downward load).
• A UDL produces a linear ramp in V and a parabolic curve in M.
• A point moment produces a jump in M.
• M is maximum where V = 0 (or jumps through zero) — this is where you read M_max for design.
• M = 0 at a free end or a pin support (no moment reaction).

3.3 Closed-form deflection — standard cases

Configuration	M_max	δ_max (location)	Notes
Simply-supported, central point load P	PL/4	PL³/(48EI) (centre)	Symmetric
Simply-supported, UDL w	wL²/8	5wL⁴/(384EI) (centre)	Most common case
Simply-supported, end moment M₀	M₀	M₀·L²/(9√3·EI) ≈ 0.0642 M₀L²/EI	Max at x = L/√3
Cantilever, tip point load P	PL (at fixed)	PL³/(3EI) (tip)	Classic robot-arm case
Cantilever, UDL w	wL²/2	wL⁴/(8EI) (tip)
Cantilever, tip moment M₀	M₀	M₀L²/(2EI) (tip)
Fixed–fixed, central point load P	PL/8	PL³/(192EI) (centre)	Stiffness 4× simply-supported
Fixed–fixed, UDL w	wL²/12 (ends)	wL⁴/(384EI) (centre)	Often governs end-moment
Propped cantilever, UDL w	wL²/8 (fixed)	wL⁴/(185EI) approx	One-step indeterminate

These six cases plus superposition cover ~80 % of textbook beam problems and a surprising fraction of real shop-floor sizing.

3.4 Reactions for common cases

Configuration	R_A	R_B	M_A (fixed)	M_B (fixed)
Simply-supported, central P	P/2	P/2	—	—
Simply-supported, P at distance a from A	P·b/L	P·a/L	—	—
Simply-supported, UDL w over full span	wL/2	wL/2	—	—
Cantilever (A fixed), tip load P	P (up)	—	PL	—
Cantilever, UDL w	wL	—	wL²/2	—
Fixed–fixed, central P	P/2	P/2	PL/8	PL/8
Fixed–fixed, UDL w	wL/2	wL/2	wL²/12	wL²/12
Propped cantilever (A fixed, B pin), UDL w	5wL/8	3wL/8	wL²/8	—
Propped cantilever, P at midspan	11P/16	5P/16	3PL/16	—

Sign convention here: reactions and moments shown as magnitudes, all acting to resist the applied load.

3.5 Worked example 1 — Steel beam, strength + serviceability

Problem. A simply-supported steel beam carries a uniform load of 5 kN/m over a 6 m span. Select a W-shape that satisfies both AISC 360-22 strength (Chapter F) and the deflection limit L/360 (live-load deflection). Use ASTM A992 (σ_y = 345 MPa = 50 ksi). E = 200 GPa.

Step 1 — Bending moment and shear.

M_max = wL²/8 = 5 · 6² / 8 = 22.5 kN·m V_max = wL/2 = 5 · 6 / 2 = 15 kN

Step 2 — Required elastic section modulus (allowable-stress check).

Using AISC ASD with φ_b → Ω_b = 1.67 (compact section), allowable bending stress F_b ≈ 0.6 σ_y = 207 MPa:

S_req = M_max / F_b = 22.5 × 10³ N·m / 207 × 10⁶ Pa = 1.087 × 10⁻⁴ m³ = 108.7 cm³ = 6.63 in³

The lightest US W-shape with S_x ≥ 6.63 in³ is W6×9 (S_x = 5.56 in³ — too small) → step to W8×10 (S_x = 7.81 in³, I_x = 30.8 in⁴ = 1.282 × 10⁻⁵ m⁴, weight 10 lb/ft). For metric equivalent, an IPE 160 (S_x = 109 cm³, I_x = 869 cm⁴) is the direct EU pick.

Step 3 — Deflection check (use the IPE 160, I = 869 cm⁴ = 8.69 × 10⁻⁶ m⁴).

δ_max = 5wL⁴ / (384 EI) = 5 · 5000 · 6⁴ / (384 · 200 × 10⁹ · 8.69 × 10⁻⁶) = 5 · 5000 · 1296 / (6.677 × 10⁸) = 3.24 × 10⁷ / 6.677 × 10⁸ = 48.5 × 10⁻³ m = 48.5 mm = 1.91 in

Deflection limit (typical floor, live load): L/360 = 6000 / 360 = 16.7 mm.

48.5 mm > 16.7 mm — deflection governs, not strength. This is the normal outcome for a long, lightly-loaded floor beam.

Step 4 — Re-size for deflection.

I_req = 5wL⁴ / (384 · E · δ_allow) = 5 · 5000 · 1296 / (384 · 200 × 10⁹ · 0.01667) = 3.24 × 10⁷ / 1.280 × 10⁹ = 25.3 × 10⁻⁶ m⁴ = 2530 cm⁴

Need I_x ≥ 2530 cm⁴. IPE 220 has I_x = 2772 cm⁴, S_x = 252 cm³ — picks up easily. Equivalent US shape: W8×18 (I_x = 61.9 in⁴ = 2577 cm⁴, S_x = 15.2 in³ — also passes).

Step 5 — Code summary.

Criterion	Demand	Capacity	Pass?
AISC ASD bending, F_b	M = 22.5 kN·m	M_n/Ω = 78 kN·m	✓ FoS 3.5
L/360 live-load defl	δ = 6.1 mm	δ_allow = 16.7 mm	✓
Shear V_n (Chapter G)	V = 15 kN	~250 kN (web)	✓ trivial

EC3 (EN 1993-1-1) would give a parallel result with γ_M0 = 1.0 and design moment M_Ed ≤ M_c,Rd = W_pl · f_y / γ_M0 — for a Class 1 IPE 220 with W_pl ≈ 285 cm³, M_c,Rd = 285 × 10⁻⁶ · 345 × 10⁶ = 98.3 kN·m, FoS on bending ≈ 4.4. EC3 SLS deflection limit varies by NA; UK NA recommends L/250 for total deflection, L/360 for variable.

3.6 Worked example 2 — Cantilever steel shaft

Problem. A solid 6061-T6 shaft, d = 50 mm, length L = 600 mm, cantilevered from a wall, carries a tip load P = 800 N. Find σ_max and δ_tip. Check against σ_y = 276 MPa per ASTM B221. E = 69 GPa.

Step 1 — Geometry.

I = π · d⁴ / 64 = π · (0.050)⁴ / 64 = π · 6.25 × 10⁻⁶ / 64 = 3.068 × 10⁻⁷ m⁴ S = I / c = 3.068 × 10⁻⁷ / 0.025 = 1.227 × 10⁻⁵ m³

Step 2 — Max moment.

M_max = P · L = 800 · 0.600 = 480 N·m (at the fixed end)

Step 3 — Bending stress.

σ_max = M_max / S = 480 / 1.227 × 10⁻⁵ = 39.1 × 10⁶ Pa = 39.1 MPa

Step 4 — Yield check.

n = σ_y / σ_max = 276 / 39.1 = 7.06 ✓ comfortable for static, ductile, room-temperature service. For a rotating shaft under reversed bending, fatigue governs — apply Marin factors and use Goodman/Soderberg (see fatigue-analysis).

Step 5 — Tip deflection.

δ_tip = P · L³ / (3 · E · I) = 800 · (0.600)³ / (3 · 69 × 10⁹ · 3.068 × 10⁻⁷) = 800 · 0.216 / (6.350 × 10⁴) = 172.8 / 63 500 = 2.722 × 10⁻³ m = 2.72 mm

Whether 2.72 mm is acceptable depends on the application — for a machine-tool cantilever it’s grossly too much (microns matter); for a pickup-truck mirror arm it’s fine.

3.7 Worked example 3 — Unit-load (Mohr–Maxwell) deflection at off-centre point

Problem. A simply-supported beam of length L carries a single point load P at x = a (where a < L/2). Find the deflection at an arbitrary point x = ξ using the unit-load method (Mohr–Maxwell virtual-work integral).

Theory (Mohr–Maxwell). The deflection at point B in the direction of a unit virtual force is:

δ_B = ∫₀ᴸ  m(x) · M(x) / (E·I) dx

where M(x) is the real bending-moment diagram and m(x) is the bending-moment diagram from a unit virtual load placed at B in the direction of the desired deflection.

Step 1 — Real moment diagram for P at x = a.

Reactions: R_A = P(L−a)/L, R_B = Pa/L.

M(x) = R_A · x = P·(L−a)·x / L           for 0 ≤ x ≤ a
M(x) = R_A · x − P·(x − a)               for a ≤ x ≤ L
     = P·a·(L − x) / L

Step 2 — Virtual moment from unit load at x = ξ (assume ξ < a).

m(x) = (L − ξ)·x / L                     for 0 ≤ x ≤ ξ
m(x) = ξ·(L − x) / L                     for ξ ≤ x ≤ L

Step 3 — Integrate piece-wise over [0, ξ], [ξ, a], [a, L].

δ(ξ) = (1/EI) · [
        ∫₀^ξ  (L−ξ)x/L · P(L−a)x/L  dx
      + ∫_ξ^a ξ(L−x)/L · P(L−a)x/L  dx
      + ∫_a^L ξ(L−x)/L · Pa(L−x)/L  dx
       ]

After integration and algebra (Hibbeler Ch. 14 walks through this), the closed form for ξ < a is:

δ(ξ) = P·(L−a)·ξ · [ L² − (L−a)² − ξ² ] / (6 · L · EI)

For a = L/2 (load at midspan), and ξ = L/2, this collapses to the familiar PL³/(48EI). The Mohr–Maxwell integral is the workhorse for any elastic-deflection problem where a closed form isn’t tabulated — including curved members, indeterminate beams (via the force method), and frame structures.

3.8 Section-property table

Useful properties for common cross-sections. y_c = centroid from a reference edge; I = second moment about the centroidal axis perpendicular to bending; S = I/c.

Section	A	I (about centroidal NA)	c (extreme fibre)	S = I/c
Rectangle b × h	b·h	b·h³/12	h/2	b·h²/6
Hollow rect b_o×h_o, b_i×h_i (concentric)	b_o·h_o − b_i·h_i	(b_o·h_o³ − b_i·h_i³)/12	h_o/2	(b_o·h_o³ − b_i·h_i³)/(6·h_o)
Solid circle, dia d	π·d²/4	π·d⁴/64	d/2	π·d³/32
Hollow circle d_o, d_i	π(d_o²−d_i²)/4	π(d_o⁴ − d_i⁴)/64	d_o/2	π(d_o⁴ − d_i⁴)/(32·d_o)
I-beam (approx, thin flanges)	2·b_f·t_f + h_w·t_w	b_f·t_f·d²/2 + t_w·h_w³/12 (d = c-to-c flanges)	h_total/2	I/c
T-section (b × h flange, t_w × h_w web)	b·t_f + h_w·t_w	parallel-axis, two rectangles	depends — find ȳ first	I/c
Triangle base b, height h	b·h/2	b·h³/36	2h/3 (apex up)	b·h²/24

For an I-beam, the exact I includes web–flange fillets and is given in AISC/EN profile tables. The approximation above is within ~3 % for typical W-shapes.

4. Reference data

4.1 US W-shapes (AISC 15th ed., abbreviated)

Shape	d (mm)	b_f (mm)	A (cm²)	I_x (cm⁴)	S_x (cm³)	Z_x (cm³)	weight (kg/m)
W6×9	150	100	17.1	928	124	142	13.4
W8×10	200	100	19.0	1282	128	147	14.9
W8×18	207	133	34.2	2577	249	281	26.8
W10×22	258	146	41.9	4954	384	432	32.7
W12×26	310	165	49.4	8492	547	612	38.7
W14×30	352	171	56.8	12 070	686	770	44.6
W18×35	449	152	66.5	21 060	938	1058	52.1
W21×50	528	166	94.8	40 790	1547	1733	74.5
W24×62	603	178	117.4	64 100	2130	2392	92.3
W30×99	753	266	187.7	168 000	4458	4998	147.3
W36×302	950	422	572.0	856 500	18 030	20 240	449.6

Source: AISC Steel Construction Manual, 15th ed., 2017, Table 1-1. Cited to two-three significant figures; consult the Manual for full precision. ASTM A992 is the default material (σ_y = 345 MPa, σ_u = 450 MPa) per AISC default since 2001.

4.2 EU IPE / HEA / HEB sections (selected, EN 10025-2 / S275 or S355)

Shape	h (mm)	b (mm)	A (cm²)	I_y (cm⁴)	W_el,y (cm³)	W_pl,y (cm³)	mass (kg/m)
IPE 100	100	55	10.3	171	34.2	39.4	8.10
IPE 160	160	82	20.1	869	109	124	15.8
IPE 220	220	110	33.4	2772	252	285	26.2
IPE 300	300	150	53.8	8356	557	628	42.2
IPE 400	400	180	84.5	23 130	1156	1307	66.3
IPE 500	500	200	116.0	48 200	1928	2194	90.7
HEA 200	190	200	53.8	3692	388.6	429.5	42.3
HEA 300	290	300	112.5	18 260	1259	1383	88.3
HEB 200	200	200	78.1	5696	569.6	642.5	61.3
HEB 300	300	300	149.1	25 170	1678	1869	117.0

Source: EN 10365:2017 Hot rolled steel channels, I and H sections — Dimensions and masses. Plastic modulus W_pl is used in EC3 ULS bending check.

4.3 Channels and hollow structural sections

C-channels (US, AISC Manual): C6×8.2 (S_x = 22.4 cm³), C8×11.5 (S_x = 53.1 cm³), C10×15.3 (S_x = 92.5 cm³), C12×20.7 (S_x = 145 cm³), C15×33.9 (S_x = 271 cm³).

HSS rectangular (cold-formed, ASTM A500 Gr. C, σ_y = 345 MPa): HSS4×4×1/4 (S_x = 56.4 cm³), HSS6×6×3/8 (S_x = 191 cm³), HSS8×8×1/2 (S_x = 466 cm³), HSS12×12×5/8 (S_x = 1320 cm³).

HSS round (ASTM A500 Gr. C / EN 10219 S355): pipe-section behaviour with high torsional and biaxial bending efficiency.

4.4 Material allowable stresses for common steels and aluminums

Material	σ_y (MPa)	σ_u (MPa)	Code allowable bending stress (AISC ASD, F_b ≈ 0.6 σ_y)
A36	250	400–550	150 MPa
A572 Gr. 50	345	450	207 MPa
A992	345	450	207 MPa
A500 Gr. C HSS	345	425	207 MPa
S235 (EN)	235	360–510	(EC3 uses partial factors, not allowable stress)
S275 (EN)	275	410–560
S355 (EN)	355	470–630
6061-T6	276	310	~165 MPa (Aluminum Design Manual ADM-2020)
2024-T3	345	483	aerospace use; Mil-Hdbk-5J / MMPDS for allowables
7075-T6	503	572	aerospace; high notch sensitivity

4.5 Deflection-limit conventions

Application	Total load	Live load only	Comment
Roof, no ceiling	L/180	L/120	IBC Table 1604.3
Roof, ceiling not susceptible	L/240	L/180
Floor, plaster or drywall ceiling	L/240	L/360	”L/360” is the canonical floor live-load limit
Industrial / craneway runways	L/600 to L/800	—	Steel-mill cranes per AIST TR-6
Cantilever (multiplier on L)	2L/240 etc.	2L/360 etc.	Code limits use the cantilever span doubled
Vibration / floor “feel”	natural freq > 8 Hz	—	AISC Design Guide 11 (Murray et al.)

In the EU, EN 1993-1-1 §7.2 + national annex give parallel limits; UK NA recommends total δ ≤ L/250, variable δ ≤ L/360 for floors.

5p. Theory

Derivation of the Euler–Bernoulli equation

Start from three ingredients:

1. Kinematics (compatibility). Plane sections remain plane and ⊥ to the deformed neutral axis:

u(x, y) = − y · dy/dx           (axial displacement of fibre at distance y)
ε_xx = ∂u/∂x = − y · d²y/dx²

2. Constitutive law (Hooke’s).

σ_xx = E · ε_xx = − E · y · d²y/dx²

3. Equilibrium. Define internal moment about the NA:

M = − ∫_A σ_xx · y dA = ∫_A E · y² · (d²y/dx²) dA = E · I · d²y/dx²

so M = EI · κ. Differentiating equilibrium of an infinitesimal slice (force balance and moment balance, with q(x) the distributed transverse load):

dV/dx = − q(x)            (force balance)
dM/dx = V                  (moment balance, neglecting V·dx · (small) term)

Substituting M = EI · d²y/dx² and differentiating twice:

d²M/dx² = EI · d⁴y/dx⁴ = dV/dx = − dV/dx (with sign convention) → EI · d⁴y/dx⁴ = q(x)

That’s the Euler–Bernoulli beam equation in its load form. Signs flip depending on whether you take q positive upward or downward; pick one and stick with it.

Timoshenko–Ehrenfest extension

Drop assumption (2) (plane sections normal to NA), keep (1) (plane). Define ψ(x) as the rotation of the cross-section, independent of dy/dx. Shear strain:

γ = dy/dx − ψ                    (Timoshenko kinematic relation)
V = k · G · A · γ = k · G · A · (dy/dx − ψ)
M = E · I · dψ/dx

The pair of governing equations (above) reduces to a single fourth-order PDE in steady-state, but separately tracks rotary inertia in dynamics — important above the first few modes.

Why k (shear-correction factor)? Jourawski shows τ is not uniform across the section: τ_max ≈ 1.5 · V/A for a rectangle, while the energy-equivalent uniform stress is V/(k·A) with k = 5/6. Cowper (1966) tabulated more precise values: 5/6 rectangle, 0.9 thin-walled I (web-dominated), 0.5 thin tube. Modern FEA codes often use 5/6 as a generic default for solid sections.

Curved-beam corrections (Winkler–Bach)

For an initially curved beam with mean radius R and depth h, the neutral axis shifts away from the centroid by e toward the centre of curvature:

σ = M · y / (A · e · (R_n − y))

with R_n = neutral-axis radius and e = R_centroid − R_n. For R/h < 5 (crane hooks, chain links, curved beams in machine frames), the curvature correction is significant — the inside fibre stress is higher than straight-beam theory predicts; the outside fibre stress is lower.

Composite beams — transformed-section method

For a beam of two materials bonded so they share strain at the interface (e.g. steel-reinforced concrete, wood with steel cover plates, sandwich panels), define the modular ratio:

n = E_2 / E_1

and transform the area of material 2 by widening it by factor n (or narrowing it, depending on which is the reference). The transformed section is analysed by ordinary beam theory: σ in material 1 from σ = My/I; σ in material 2 by multiplying the transformed-section result by n.

For prestressed and reinforced concrete (RC), the same idea is used with n = E_steel / E_concrete ≈ 6–10, and cracked-section analysis (ignoring tension in concrete below the NA) is the working method. See reinforced-concrete.

6p. Application

Sizing — strength vs deflection trade-off

Two governing conditions, computed independently:

1. Strength: S_req,strength = M_max / σ_allow
2. Stiffness: I_req,defl = (load coefficient) · w · L⁴ / (E · δ_allow)

For a simply-supported UDL, the deflection requirement gives I_req = 5wL⁴ / (384 · E · δ_allow). The actual S of the required section may exceed S_req,strength by a huge multiple — that’s the normal outcome for floors and is the reason building floors aren’t designed to AISC strength alone.

A practical heuristic: when the span exceeds ~20–25× the depth, deflection governs; when shorter, strength governs. Roof beams (deflection limit L/240, lower live load) more often hit strength; floor beams (L/360, high live load) more often hit deflection.

Reading code tables

• AISC Steel Construction Manual Part 3 lists beam selection tables (Z_x tables) sorted by plastic modulus. Pick the lightest shape whose Z_x ≥ M_u / (φ·F_y); then check serviceability separately.
• EN 1993-1-1 uses the same logic with W_pl,y instead of Z_x and partial factor γ_M0 = 1.0 instead of φ. Member buckling (lateral-torsional, LTB) is the Chapter 6.3 check that often controls.
• Aluminum Design Manual (ADM) Part 1 provides parallel allowables for 6061-T6, 6063-T5, etc.

Beam taxonomy in construction

The construction trades use very specific role-based beam names — knowing which is which prevents miscommunication on shop drawings:

• Girder — a primary beam that supports secondary beams (joists, purlins). Heaviest section.
• Joist — closely spaced (typically 300–600 mm OC for wood, 600–1500 mm for open-web steel) beam supporting floor or roof sheathing.
• Stringer — bridge longitudinal beam; or stair carriage; or wood-frame rim board.
• Purlin — horizontal roof beam spanning between primary frames, supporting roof deck.
• Header — beam over an opening (door, window) transferring load around the void.
• Lintel — masonry equivalent of a header.
• Rafter — sloped roof beam spanning eave to ridge.
• Spandrel beam — perimeter beam supporting exterior cladding plus floor load.

W vs HSS vs C — selection

• W (wide-flange) — most efficient bending shape per unit weight; the default. Weak in torsion (open section, low J).
• HSS (hollow structural section, rectangular or round) — much higher torsional stiffness (closed section, J orders of magnitude larger). Better for columns, biaxial bending, and exposed architectural members. Slightly more expensive per kg.
• C (channel) — singly-symmetric. Centroid and shear centre are separated → transverse loads through the centroid produce both bending and torsion. Use mostly for runways, edge stiffeners, secondary framing. Avoid as primary bending member unless braced.
• L (angle) — used in trusses, bracing, and built-up shapes. Rarely a pure bending member.
• Plate girder (welded I-shape, custom dimensions) — for spans / loads beyond rolled-shape availability (bridges, industrial buildings with > ~30 m spans).

7p. Edge cases & assumptions

1. Deep beams (L/h < ~10). Euler–Bernoulli underestimates deflection. Use Timoshenko, or for very deep beams (L/h < 4) treat as a 2-D plane-stress problem in FEA — the simple beam idealization no longer captures the load path. ACI 318 / EC2 define a deep concrete beam as L_n/h ≤ 4 and require strut-and-tie design.

2. Saint-Venant violation near supports and concentrated loads. Within roughly one cross-section depth of a support or point load, the actual stress distribution is 3-D and not given by σ = My/I. AISC web-yielding/web-crippling checks (Chapter J10) exist exactly to catch this for steel beams; concrete bearing-stress provisions (ACI 318 §22.8) handle it for RC.

3. Large deflections. When δ exceeds a few percent of L, the small-slope assumption fails. Use the elastica solution (Euler 1744), or geometric-nonlinear FEA. Common in flexible-arm robotics, fishing-rod design, snap-fit features.

4. Lateral-torsional buckling (LTB). An unbraced compression flange can buckle sideways before the section reaches its full plastic moment. AISC Chapter F2.2 modifies the bending capacity by a factor C_b · F_cr that depends on the unbraced length L_b and the moment gradient. Bracing intervals shorter than L_p (plastic length) eliminate LTB; longer than L_r (elastic LTB limit) drops to fully elastic buckling. Same idea in EC3 §6.3.2 with the χ_LT reduction factor.

5. Local buckling of compression flanges and webs. Slender plates inside the section can buckle before the gross section yields. Slenderness limits (b/t for flanges, h/t_w for webs) classify sections as Class 1 (compact, plastic), Class 2 (compact, plastic with reduced rotation), Class 3 (semi-compact, elastic), or Class 4 (slender, post-buckling effective section). EC3 §5.5, AISC B4.

6. Composite action with concrete deck. A steel beam with shear studs into a concrete slab behaves as a composite beam, with an effective slab width b_eff. AISC Chapter I and EC4 (EN 1994-1-1) provide b_eff formulas (typically L/8 each side of beam, limited by beam spacing). Composite action can double the elastic stiffness and increase moment capacity by 30–60 %.

7. Shear deformation in short cantilevers. Robot links, machine-tool tooling cantilevers, and 3-D-printed brackets often have L/h ≈ 3–5. E–B underestimates δ_tip by 10–30 %. Use Timoshenko or check with FEA solid elements. Closed-form Timoshenko cantilever-tip deflection under tip load P:

δ_tip = P·L³/(3EI) + P·L/(k·G·A)
        ↑ E–B bending     ↑ shear correction

1. Shear-locking in FEA. Linear-Lagrange Timoshenko beam elements over-stiffen unless reduced integration or assumed-strain methods are used. Hermite-cubic E–B beams don’t lock but exclude shear. ABAQUS B21/B31 (Timoshenko) and B23/B33 (E–B) — choose deliberately.

2. Anisotropic / orthotropic beams. Wood (orthotropic), CFRP laminates, and additively-manufactured FDM beams violate the isotropic Hooke’s-law step. For laminates, classical lamination theory (CLT) gives an effective EI; for wood, use E_∥ (parallel to grain) and apply the volume / load-duration / moisture adjustment factors of NDS or EC5.

3. Plastic hinge formation and post-yield behaviour. Steel beams beyond M_y form a plastic zone that, in compact sections, fully plasticizes the section at M_p = Z · σ_y > M_y. M_p / M_y is the shape factor ≈ 1.5 for solid rectangles, ~1.12–1.18 for W-shapes, ~1.7 for solid round bars. Plastic design (AISC Appendix 1, EC3 §5.4.3) uses M_p and limits redistribution by the rotation capacity.

4. Vibration coupling. A beam’s first natural frequency under self-weight is f₁ ≈ (1/2π) · √(g / δ_static) for many configurations (Rayleigh approximation). For floors, AISC Design Guide 11 sets a “feel” target of f₁ ≥ ~8 Hz to avoid pedestrian-induced annoyance. See vibration-dynamics.

5. Temperature gradients. A through-thickness ΔT in a constrained beam induces both axial force and bending moment. Bridges, refractory linings, and aerospace heat-shielded structures handle this routinely (expansion joints, sliding bearings, composite layups with tuned CTEs).

8p. Tools & software

Hand-calc tools

• AISC Steel Construction Manual, 15th ed. (2017, ANSI/AISC 360-16; 16th ed. 2023 with AISC 360-22) — the bible for US steel beam design; Z_x tables, beam-curve charts, K-factors, bolt and weld design.
• Eurocode 3 (EN 1993-1-1:2022) plus the relevant National Annex — EU equivalent; companion EN 10365 profile tables (IPE, HEA, HEB, UPN).
• Roark’s Formulas for Stress and Strain, 8th ed. (Young, Budynas, Sadegh, 2020) — every closed-form beam case ever derived, with deflections and stresses for curved beams, plates, shells. The single most-cited engineering reference.
• Aluminum Design Manual (ADM-2020) — Aluminum Association allowables and beam-design provisions for 6xxx and 7xxx alloys.

Structural analysis software

• RISA-2D / RISA-3D — Mid-market structural-analysis package, intuitive UI, AISC and EC3 design-checks built in.
• SkyCiv — Browser-based; beam, truss, frame, 3-D FEA, design checks. Strong for small-firm and educational use.
• SAP2000 (CSI) — Industry-standard for buildings and bridges; integrated steel/concrete/aluminum design.
• ETABS (CSI) — Building-focused variant; floor-system analysis, composite beam design.
• STAAD.Pro (Bentley) — Heavyweight in industrial and infrastructure; multi-code design (AISC, EC, IS, BS, AS, JIS).
• Robot Structural Analysis (Autodesk) — Integrated with Revit; common in EU markets.
• ConSteel — Strong on LTB and member stability under EC3.

General-purpose FEA for beam problems

• ANSYS Mechanical — BEAM188 (Timoshenko, 3-D linear-quadratic) is the workhorse element.
• Abaqus — B21/B31 (Timoshenko 2-D/3-D), B23/B33 (cubic E–B), B22/B32 (quadratic Timoshenko).
• NASTRAN — CBAR / CBEAM elements (Timoshenko-equivalent shear-flexible).
• CalculiX (open-source, Abaqus-compatible) — B31, B32 beams.
• OpenSees (open-source) — forceBeamColumn, dispBeamColumn elements with fiber sections; widely used in earthquake-engineering research.
• CodeAster (open-source, EDF) — heavy on nonlinear; full 3-D-beam capability with warping (Vlasov).

Spreadsheet / parametric

• Excel / LibreOffice + iterative tables — most fabrication shops still size routine beams in Excel, often built around an AISC Z_x lookup.
• MathCAD / SMath / Jupyter notebooks — for one-off custom configurations (Mohr–Maxwell integrals, indeterminate beams via force method, fatigue checks).
• MATLAB / Octave / Python (SymPy) — for derivations and parameter studies; useful for generating shear-and-moment diagrams of arbitrary load combinations.

Open-source niche

• Concrete (a Python package for RC design) — implements ACI 318 / EC2 cracked-section analysis.
• anaStruct (Python) — 2-D frame solver, free, scriptable for parametric studies.
• PyNiteFEA (Python) — 3-D frame FEA with AISC steel-design module.
• Frame3DD — small, scriptable 3-D frame solver in C; embeds well in design pipelines.

11. Cross-references

• mechanics-of-materials — provides the stress, strain, Hooke’s-law foundation that beam theory specializes.
• statics-fundamentals — reactions and load determination before internal forces.
• structural-analysis — extends beam theory to indeterminate systems (force method, displacement method, moment distribution, stiffness matrix).
• steel-design — applies E–B + LTB + local-buckling provisions to steel members per AISC 360-22 and EC3.
• reinforced-concrete — RC beam design uses transformed-section / cracked-section ideas built on the same kinematic foundation.
• fatigue-analysis — cyclic-loading extension; uses M_max from beam theory in Goodman / Soderberg checks.
• vibration-dynamics — beam vibration via E–B EOM EI · y'''' + ρA · ÿ = q(x, t); modal analysis.
• materials-steel, materials-aluminum — material allowables for σ_y, σ_u, E.
• fasteners-bolts — bolted moment connections; bolt design under combined moment + shear.
• pipe-fittings — head bending and saddle stresses are beam-on-elastic-foundation problems.
• manipulator-design — robot links analysed as cantilever beams; tip deflection is a primary repeatability metric. (To be written.)
• scientific — beam-element keywords in .inp (Abaqus), .bdf (NASTRAN), .cdb (ANSYS).

12. Citations

1. Hibbeler, R. C. Mechanics of Materials, 11th ed. Pearson, 2022. ISBN 978-0137605446. Chapters 6–9, 12: bending stress, shear flow, deflection, indeterminate beams.
2. Beer, F. P.; Johnston, E. R.; DeWolf, J. T.; Mazurek, D. F. Mechanics of Materials, 8th ed. McGraw-Hill, 2019. ISBN 978-1260113273. Chapters 4–6, 9: bending, shear in beams, deflection.
3. Gere, J. M.; Goodno, B. J. Mechanics of Materials, 9th ed. Cengage, 2018. ISBN 978-1337093347. Chapters 5–6, 9–10: pure and non-uniform bending, shear stress, deflection by integration and superposition.
4. Timoshenko, S. P.; Gere, J. M. Mechanics of Materials, 2nd ed. PWS, 1972. ISBN 978-0534921743. Classic source for both E–B and Timoshenko theory by the latter’s eponym.
5. Timoshenko, S. P.; Goodier, J. N. Theory of Elasticity, 3rd ed. McGraw-Hill, 1970. ISBN 978-0070858053. Foundational elasticity theory; underpins all beam-theory derivations.
6. Young, W. C.; Budynas, R. G.; Sadegh, A. M. Roark’s Formulas for Stress and Strain, 8th ed. McGraw-Hill, 2020. ISBN 978-1260453751. Tables 8.1–8.10: deflections, slopes, reactions, moments for every common beam case. The single most-cited reference book in stress analysis.
7. Cowper, G. R. “The Shear Coefficient in Timoshenko’s Beam Theory.” Journal of Applied Mechanics, vol. 33, no. 2, 1966, pp. 335–340. doi:10.1115/1.3625046. The reference paper for shear-correction factor k for arbitrary cross-sections.
8. Shigley, J. E.; Budynas, R. G.; Nisbett, J. K. Shigley’s Mechanical Engineering Design, 11th ed. McGraw-Hill, 2020. ISBN 978-0073398211. Chapter 4: deflection and stiffness; Chapter 7: shaft design with combined bending and torsion.
9. McCormac, J. C.; Csernak, S. F. Structural Steel Design, 6th ed. Pearson, 2018. ISBN 978-0134589657. AISC 360-based steel beam design, including LTB and Cb modification.
10. Salmon, C. G.; Johnson, J. E.; Malhas, F. A. Steel Structures: Design and Behavior, 5th ed. Pearson, 2008. ISBN 978-0131885561. Rigorous text on AISC steel beam and beam-column design.
11. AISC 360-22 — Specification for Structural Steel Buildings. American Institute of Steel Construction, 2022. Chapter F: Design of members for flexure; Chapter G: shear; Chapter H: combined forces.
12. AISC Steel Construction Manual, 15th ed., 2017 (16th ed., 2023). Beam selection tables (Z_x, S_x), shape properties, beam-design charts.
13. EN 1993-1-1:2022 — Eurocode 3: Design of steel structures — Part 1-1: General rules and rules for buildings. CEN, 2022. §5–6 cover bending, shear, and stability of beams.
14. EN 1994-1-1:2004 — Eurocode 4: Design of composite steel and concrete structures — Part 1-1. Composite beam design (steel + concrete deck with shear connectors).
15. ASTM A992/A992M-22 — Standard Specification for Structural Steel Shapes (the default σ_y = 50 ksi grade for US W-shapes). ASTM International, 2022.
16. ASTM E8/E8M-22 — Standard Test Methods for Tension Testing of Metallic Materials. The reference for σ_y, σ_u, E used in every beam allowable.
17. Murray, T. M.; Allen, D. E.; Ungar, E. E.; Davis, D. B. AISC Design Guide 11: Vibrations of Steel-Framed Structural Systems Due to Human Activity, 2nd ed. AISC, 2016. The standard for floor-vibration serviceability checks.