Centrally-Clamped Parallel-Beam Bistable Calculator

Mechanism type

envelope constraint

width (mm) ← side view horiz

height (mm) ← side view vert

depth (mm) ← top view

flexure

t flexure thickness (mm)

L_fi inner beam length (mm)

L_fo outer beam length (mm)

geometry

L_s rigid segment (mm)

θ initial angle (°)

central shuttle

central shuttle width (mm)

stacking

N arm pairs

1 pairs

s arm spacing (mm)

material

preset

E Young's modulus (MPa)

σ_y yield strength (MPa)

σ_fat fatigue limit (MPa)

side view — snap-plane cross-section

top view — depth & stacking

key outputs

bistability ratio BR

—

Snap force ↓to / ↑back

—

PRBM peak stress

—

k_oop / k_ip

—

× lateral stiffness ratio

Q parameter

—

Stroke (d_travel)

—

total snap displacement

Max strain ε

—

Qiu Eq. 46: 2π²th/l²

2nd stable pos. (d_end)

—

Qiu Eq. 44 · 1.99 × H

Snap-through at (d_top)

—

Qiu Eq. 44 · 0.16 × H

Mid-travel (d_mid)

—

Qiu Eq. 44 · 4/3 × H

Force asymmetry r

—

Qiu Eq. 48 · F_top / F_bot

Min beam length (l_min)

—

Qiu Eq. 47

FEA analysis

FEA is not yet available. Contact me at dbalogh@udel.edu for questions.

total combinations

—

bistable combos

—

fatigue safe

—

snap force range (N)

—

snap force distribution — bistable combinations only

all combinations

t	θ°	Ls	I	Kt	Kt_fillet	H	BR	bistable	F_snap (N)	σ_inner (MPa)	σ_outer (MPa)	σ_peak (MPa)	SF	fatigue	k_ip	k_oop	oop/ip

parameter locks

locked = fixed at current design value · sweep = optimizer finds best value across full slider range

flexure

t —

Lf_inner —

Lf_outer —

Ls —

θ —

geometry

central shuttle width—

stacking

N (pairs) —

arm spacing—

hard constraints — disqualifies combinations before scoring

minimum safety factor SF = yield ÷ peak stress · below this the flexure yields and the mechanism breaks 1.5

target safety factor scoring reward plateaus once SF reaches this value — no benefit to over-engineering beyond it 2.5

scoring weights — applied only to designs that pass the SF floor

snap force (higher = better)5

Q (bistability) (higher = better)8

stress margin — reward reaching target SF, plateau beyond it prevents optimizer always pushing to minimum theta · lower stress = more cycles before fatigue failure6

top 5 combinations ★ = optimal (swept) · = locked

all valid combinations — ranked (bistable & SF ≥ minimum)

rank	score	t	Lf_i	Lf_o	Ls	θ°	N	BR	F_snap (N)	σ (MPa)	SF	fatigue

checks stress at both beam roots across the full snap-through travel (+θ through 0° to −θ). each beam independently deflects by the full arm angle θ — the shorter beam is always more stressed. beam lengths Lf_inner and Lf_outer are taken from the design tab.

stress at both beam roots across full travel

position A (+θ) — stable

inner beam σ (at post)—

outer beam σ (at wall)—

peak stress—

position B (θ/2) — mid travel

inner beam σ (at post)—

outer beam σ (at wall)—

peak stress—

position C (0°) — through zero

inner beam σ (at post)—

outer beam σ (at wall)—

peak stress—

position D (−θ) — stable

inner beam σ (at post)—

outer beam σ (at wall)—

peak stress—

stress vs travel angle — all 4 beam ends

material comparison — same geometry

material	E (MPa)	yield (MPa)	fatigue (MPa)	peak stress (MPa)	yield margin	fatigue SF	survives snap?	fatigue safe?

overall verdict

stress map — all critical points

beam lengths from design tab: Lf_inner = 3.64 mm, Lf_outer = 4.01 mm. fatigue limit = 25 MPa · yield = 50 MPa.

location	beam	L_nom (mm)	L_eff (mm)	pos A (+θ)	pos B (θ/2)	pos C (0°)	pos D (−θ)	peak σ (MPa)	% of yield	% of fatigue	status

derived values — current design

all values computed from the current design tab settings. units: mm, N, MPa.

derived geometry

Lf_total (nominal)—

inner beam L_eff—

outer beam L_eff—

Lf_eff_total—

fillet radius (fixed)R = 0.2 mm

Kt_fillet @ current t—

stiffness & material @ current design

in-plane k_ip—

out-of-plane k_oop—

k_oop / k_ip—

shear modulus G—

arm material mass—

fatigue SF threshold1.5×

fatigue limit basis½ × yield (Slocum/MIT)

equations — pseudo-rigid body model (PRBM)

all dimensions in mm · forces in N · stress in MPa · source: Howell (2001) Compliant Mechanisms; Zirbel et al. (2016) PLoS ONE

PRBM constants (Howell 2001, Table 5-2)

char. spring constantK_Θ = 2.67617

char. radius factorγ = 0.8517

fillet radius (fixed)R = 0.2 mm

K_Θ and γ are derived from energy minimisation of the PRBM cantilever deflection curve. They are universal constants for a tip-loaded cantilever and apply directly to each compliant segment.

geometry & cross-section

moment of inertiaI = depth · t³ / 12

geometric flex lengthL_flex = L_nominal − 2R (fillet correction)

total flex lengthLf_eff = Lfi_flex + Lfo_flex

fillet stress conc.Kt_fillet = 1 + 0.5 · √(t / 2R)

vertical offset HH = Lf_eff · sin(θ) (flex segments only)

torsional stiffness & snap force

Kt per beamKt_i = K_Θ·EI/Lfi_eff Kt_o = K_Θ·EI/Lfo_eff

total Kt (parallel)Kt = Kt_i + Kt_o = K_Θ·EI·(Lfi+Lfo)/(Lfi·Lfo)

both beams deflect independently by θ → torsional stiffnesses add (parallel, not series)

bistability ratioBR = H / (t · √3)

bistable conditionBR > 1 (Zirbel et al. 2016, Table 4)

snap forceF = N · 2Kt · θ / (Ls · cos θ)

F, k_ip, k_oop all scale linearly with N arm pairs. snap force is the peak at departure from stable state 1.

stress — independent cantilever PRBM

inner beam root σσ_i = K_Θ · E · t · θ / (2 · Lfi_eff) · Kt_fillet

outer beam root σσ_o = K_Θ · E · t · θ / (2 · Lfo_eff) · Kt_fillet

peak stressσ = max(σ_i, σ_o) — shorter beam is more stressed

fatigue SFSF = σ_fatigue / σ_peak (safe if SF ≥ 1.5)

Each beam is a cantilever independently deflecting by the full arm angle θ. The shorter beam bends more sharply for the same angle → higher root stress. Source: Howell (2001) PRBM cantilever stress formula applied per beam.

stiffness — corrected series formula

in-plane k_ip2N · E · depth · t³ / (Lfi³ + Lfo³)

out-of-plane k_oop2N · E · depth³ · t / (Lfi³ + Lfo³)

Series flex beams: 1/k = Lfi³/(EI) + Lfo³/(EI). Using (Lfi+Lfo)³ is only valid when Lfi=Lfo; for unequal lengths it overestimates stiffness by up to 4×.

k_oop/k_ip = (depth/t)². large depth → very stiff out-of-plane, prevents lateral buckling.

material-derived quantities

shear modulusG = E / (2·(1 + ν))

arm material massm = ρ · V_arms / 1000 (g, ρ in g/cm³)

arm volumeV = 2N · [(Lfi+Lfo)·t + Ls·c] · depth (mm³)

ν does not appear in the PRBM bending equations directly — it only affects G, which matters if torsional (twisting) vibration modes are considered. mass covers flex and rigid arm segments only; post and end blocks are excluded.

what changed vs previous version

corrected (this version)

Kt: added K_Θ=2.676 factor → F_snap ~2.7× larger

σ: uses Lf_eff_total not individual Lfi/Lfo

σ: inner = outer (series moment sharing)

k_ip, k_oop: denominator Lfi³+Lfo³ not (Lfi+Lfo)³

G = E/(2(1+ν)) now computed from ν

mass = ρ·V_arms/1000 now computed from ρ

unchanged & confirmed correct

BR = H/(t·√3) bistability condition

H = Lf_eff · sin(θ)

I = depth · t³ / 12

Kt_fillet = 1 + 0.5·√(t/2R)

model assumptions & known limitations

included in model

✓ PRBM torsional stiffness with K_Θ = 2.676

✓ series flex: correct Lfi³+Lfo³ stiffness

✓ series stress: equal σ at inner & outer roots

✓ fillet stress concentration (Kt_fillet)

✓ N arm-pair force/stiffness scaling

✓ in-plane & out-of-plane stiffness

✓ envelope constraint checking

not modeled (limitations)

✗ large-angle correction (θ > 15° loses accuracy)

✗ FDM anisotropy (E can be 20–40% lower in Z)

✗ post-buckling force curve (only peak force)

✗ creep / relaxation (polymer fatigue overestimated)

✗ rigid segment inertia (dynamic snap speed)

✗ contact / end-stop behaviour