Roark's Formulas for Stress and Strain
Shear, moment, slope, deflection of elastic straight beams-table 8.1
W=load (N), w=unit load (N/mm), M0=applied couple (Nmm), θ0=angular displacement (rad),
RA and RB are the vertical end reactions at left and right respectively (N),
MA and MB are the reaction end moments at left and right respectively (Nmm)
2. Partial distributed load
partial distributed load
Unit load wa (N/mm): 
Unit load wl (N/mm): 
Distance l (mm): 
Distance a (mm): 
Elastic modulus E (MPa): 
Moment of Inertia I (Nmm4): 
Distance neutral axis to outer fiber v (mm): 
 
2a. Left end free, right end fixed (cantilever)
load case 1a
RA (N): RA RB (N): RB
MA (Nmm): MA MB (Nmm): MB
σA (MPa): σA σB (MPa): σB
θA (degrees): θA θB (degrees): θB
deflection yA (mm): yA deflection yB (mm): yB
If a=0 and wl = wa (uniform load on the entire span)
Max M = MB (Nmm): maxM
Max σ = σB (MPa): maxσ
Max θ = θA (degrees): Maxθ
Max y = yA (mm): maxY
If a=0 and wa = 0 (uniformly increasing load)
Max M = MB (Nmm): maxM
Max σ = σB (MPa): maxσ
Max θ = θA (degrees): Maxθ
Max y = yA (mm): maxY
If a=0 and wl = 0 (uniformly decreasing load)
Max M = MB (Nmm): maxM
Max σ = σB (MPa): maxsigma
Max θ = θA (degrees): Maxθ
Max y = yA (mm) maxY
2b. Left end guided, right end fixed
load case 2b
RA (N): RA RB (N): RB
MA (Nmm): MA MB (Nmm): MB
σA (MPa): σA σB (MPa): σB
θA (degrees): θA θB (degrees): θB
deflection yA (mm): yA deflection yB (mm): yB
If a=0 and wl = wa (uniform load on the entire span)
Max -M = MB (Nmm): max-M
Max -σ = σB (MPa): max-σ
Max +M = MA (Nmm): max+M
Max +σ = σA (MPa): max+σ
Max y = yA (mm): maxY
If a=0 and wa = 0 (uniformly increasing load)
Max -M = MB (Nmm): max-M
Max -σ = σB (MPa): max-σ
Max +M = MA (Nmm): max+M
Max +σ = σA (MPa): max+σ
Max y = yA (mm): maxY
If a=0 and wl = 0 (uniformly increasing load)
Max -M = MB (Nmm): max-M
Max -σ = σB (MPa): max-σ
Max +M = MA (Nmm): max+M
Max +σ = σA (MPa): max+σ
Max y = yA (mm): maxY
2c. Left end simply supported, right end fixed
load case 2c
RA (N): RA RB (N): RB
MA (Nmm): MA MB (Nmm): MB
σA (MPa): σA σB (MPa): σB
θA (degrees): θA θB (degrees): θB
deflection yA (mm): yA deflection yB (mm): yB
If a=0 and wl = wa (uniform load on the entire span)
RA (N): RA
RB (N): RB
Max -M = MB (Nmm): max-M
Max -σ = σB (MPa): max-σ
Max +M (Nmm): max+M at x (mm) = x
Max +σ (MPa): max+σ
Max θ = θA (degrees): maxθ
Max y (mm) maxY at x (mm) = x
If a=0 and wa = 0 (uniformly increasing load)
RA (N): RA
RB (N): RB
Max -M = MB (Nmm): max-M
Max -σ = σB (MPa): max-σ
Max +M (Nmm): max+M at x (mm) = x
Max +σ (MPa): max+σ
Max θ = θA (degrees): maxθ
Max y (mm) maxY at x (mm) = x
If a=0 and wl = 0 (uniformly decreasing load)
RA (N): RA
RB (N): RB
Max -M = MB (Nmm): max-M
Max -σ = σB (MPa): max-σ
Max +M (Nmm): max+M at x (mm) = x
Max +σ (MPa): max+σ
Max θ = θA (degrees): maxθ
Max y (mm) maxY at x (mm) = x
2d. Left end fixed, right end fixed
load case 2d
RA (N): RA RB (N): RB
MA (Nmm): MA MB (Nmm): MB
σA (MPa): σA σB (MPa): σB
θA (degrees): θA θB (degrees): θB
deflection yA (mm): yA deflection yB (mm): yB
If a=0 and wl = wa (uniform load on the entire span)
Max -M = MA=MB (Nmm): max-M
Max -σ = σA= σB (MPa): max-σ
Max +M (Nmm): max+M at x (mm) = x
Max +σ (MPa): max+σ
Max y (mm) maxY at x (mm) = x
If a=0 and wa = 0 (uniformly increasing load)
RA (N): RA
MA (Nmm): MA
σA (MPa): σA
RB (N): RB
Max -M = MB (Nmm): max-M
Max -σ = σB (MPa): max-σ
Max +M (Nmm): max+M at x (mm) = x
Max +σ (MPa): max+σ
Max y (mm) maxY at x (mm) = x
2e. Left end simply supported, right end simply supported
load case 2e
RA (N): RA RB (N): RB
MA (Nmm): MA MB (Nmm): MB
σA (MPa): sigmaA σB (MPa): σB
θA (degrees): θA θB (degrees): θB
deflection yA (mm): yA deflection yB (mm): yB
If a=0 and wl = wa (uniform load on the entire span)
RA (N): RA RB (N): RB
Max M (Nmm): maxM when x (mm) = x
Max σ (MPa): maxσ
Max θ = θB (degrees) maxθ
Max y (mm): maxY when x (mm)= x
If a=0 and wa = 0 (uniformly increasing load)
RA (N): RA RB (N): RB
Max M (Nmm): maxM when x (mm) = x
Max σ (MPa): maxσ
Max θ = θA (degrees) maxθ Max θ = θB (degrees) maxθ
Max y (mm): maxY when x (mm)= x
2f. Left end guided, right end simply supported
load case 2f
RA (N): RA RB (N): RB
MA (Nmm): MA MB (Nmm): MB
σA (MPa): σA σB (MPa): σB
θA (degrees): θA θB (degrees): θB
deflection yA (mm): yA deflection yB (mm): yB
If a=0 and wl = wa (uniform load on the entire span)
Max M = MA(Nmm): maxM
Max σ = σA(MPa): maxσ
Max θ = θB (degrees) maxθ
Max y = yA(mm): maxY
If a=0 and wa = 0 (uniformly increasing load)
Max M = MA(Nmm): maxM
Max σ = σA(MPa): maxσ
Max θ = θB (degrees) maxθ
Max y = yA(mm): maxY
If a=0 and wl = 0 (uniformly decreasing load)
Max M = MA(Nmm): maxM
Max σ = σA(MPa): maxσ
Max θ = θB (degrees) maxθ
Max y = yA(mm): maxY