Beam Deflection and Stress Equations Calculator for Beam Fixed at Both Ends, Load at any location

Structural Beam Deflection, Stress, Bending Equations and calculator for a Beam Fixed at Both Ends, Load at any location.

Beam Fixed at Both Ends, Load at any location

Beam Moment and Loads at Supports

M = Wab² / l²
W₁=(Wb² / l³⁾)(l + 2a)

Beam Section Modulus Z

Z = I/z

I = Moment of inertia Cross Section : Units in⁴ (m⁴)
z = Distance neutral axis to extreme fiber: Units in (m)

General Beam Stress at any Given Point

Beam Stress For segment length a,

Beam Stress For segment length b

For segment length b

Beam Stress at Critical Locations Points

Beam Stress at end nearest to segment of length a,

Beam Stress at end nearest to segment of length b,

Maximum Beam stress is at end nearest to shorter segment.
Beam Stress is zero at

and

Greatest negative Beam stress, at load

Beam Deflection at any Point

For length a,

For length b

Critical Beam Deflections

Beam Deflection at load

If b length is the longer segment and a is the shorter one.

The maximum Beam deflection is in the longer segment:

and

maximum deflection is in the longer segment

Where:

E =	Beam Modulus of Elasticity	psi	(N/mm²⁾
I =	Beam Moment of Inertia	in⁴	(mm⁴⁾
W =	Beam Load	lbs	(N)
s =	Beam Stress at the cross-section being evaluated	Lbs/in²	(N/mm²)
y =	Beam Deflection	inches	(mm)
x =	Some distance as indicated	inches	(mm)
u =	Some distance as indicated	inches	(mm)
a, b =	Some distance as indicated	inches	(mm)
L =	Some distance as indicated	inches	(mm)
l =	Some distance as indicated	inches	(mm)
Z =	section modulus of the cross-section of the beam = I/z	in³	(mm³⁾
z =	distance from neutral axis to extreme fiber (edge)	inches	(mm)

Please note letter "l" (lower case "L") is different than "I" (Moment of Inertia).
Deflections apply only to constant cross sections along entire length.

References: