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)
1. Concentrated intermediate load
concentrated intermediate load
Force W (N): 
Distance l (mm): 
Distance a (mm): 
Elastic modulus E (MPa): 
Moment of Inertia I (mm4): 
Distance neutral axis to outer fiber v (mm): 
 
1a. 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
Max M = MB (Nmm): maxM when a (mm) = 0
Max σ = σB (MPa): max σ    
Max θ = θA (degrees): Maxθ when a (mm) = 0
Max y = yA (mm): maxY when a (mm) = 0
1b. Left end guided, right end fixed
load case 1b
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
Max +M = MA (Nmm): max+M when a (mm) = 0
Max +σA (MPa): max+σ    
Max -M = MB (Nmm): max-M when a (mm) = 0
Max -σB (MPa): max-σ    
Max y = yA (mm): maxY when a (mm) = 0
1c. Left end simply supported, right end fixed
load case 1c
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
Max +M (Nmm): max+M at x (mm) = a
Max +σ (MPa): max+σ    
Max +M (Nmm): max+M when a (mm) = 0.366l
Max +σ (MPa): max+σ    
Max -M = MB (Nmm): max-M    
Max -σ (MPa): max-σ    
Max -M (Nmm): max-M when a (mm)= 0.5773l
Max -σ (MPa): max-σ    
Max y (mm): maxY at x (mm) = x
Max y (mm): maxY when a (mm) = 0.414l
1d. Left end fixed, right end fixed
load case 1d
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
Max +M (Nmm): max+M at x (mm) = a
Max +σ (MPa): max+σ    
Max +M (Nmm): max+M when a (mm) = l/2
Max +σ (MPa): max+σ    
Max -M=MA (Nmm): max-M if a smaller than l/2
Max -σ (MPa): max-σ    
Max -M (Nmm): max-M when a (mm)= l/3
Max -σ (MPa): max-σ    
Max y (mm): maxY at x (mm) = 2al/(l+2a) if a greater than l/2
Max y (mm): maxY when a (mm) = l/2
1e. Left end simply supported, right end simply supported
load case 1e
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
Max M (Nmm): maxMx at x (mm) = a
Max σ (MPa): max σx    
Max M (Nmm): maxM when a (mm) = l/2
Max σ (MPa): max σ    
Max y (mm): maxY at x (mm)= l-sqrt((l^2-a^2)/3) when a less than l/2
Max y (mm): maxY at x= and when a (mm)= l/2
Max θ = θA (degrees): maxθ when a less than l/2
Max θ (degrees): maxθ when a (mm) = 0.423l
1f. Left end guided, right end simply supported
load case 1f
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
Max M = MA(Nmm): maxM for x between 0 and a  
Max σ = σA(MPa): max σ    
Max M = MA(Nmm): maxM when a (mm) = 0
Max σ = σA(MPa): max σ    
Max y = yA(mm): maxY    
Max y (mm): maxY when a (mm)= 0
Max θ = θB (degrees): maxθ    
Max θ (degrees): maxθ when a (mm) = 0