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The tables below give equations for the deflection, slope, shear, and moment along straight beams for different end conditions and loadings. You can find comprehensive tables in references such as Gere , Lindeburg , and Shigley . However, the tables below cover most of the common cases.

For information on beam deflection, see our reference on stresses and deflections in beams .

Cantilever Beams
Cantilever, End Load
$$ \delta = - { Fx^2 \over 6EI } \left( 3L - x \right) $$

$$ \delta_{max} = {F L^3 \over 3EI} $$
@ x = L

$$ \theta = - { Fx \over 2EI } \left( 2L - x \right) $$

$$ \theta_{max} = {F L^2 \over 2EI} $$
@ x = L

Cantilever, Intermediate Load
$$ \delta = - { Fx^2 \over 6EI } \left( 3a - x \right) $$
( 0 ≤ x ≤ a )
$$ \delta = - { Fa^2 \over 6EI } \left( 3x - a \right) $$
( a ≤ x ≤ L )

$$ \delta_{max} = {F a^2 \over 6EI} \left( 3L - a \right) $$
@ x = L

$$ \theta = - { Fx \over 2EI } \left( 2a - x \right) $$
( 0 ≤ x ≤ a )
$$ \theta = - { Fa^2 \over 2EI } $$
( a ≤ x ≤ L )

V = +F
( 0 ≤ x ≤ a )
V = 0
( a ≤ x ≤ L )

M = −F (a − x)
( 0 ≤ x ≤ a )
M = 0
( a ≤ x ≤ L )

Cantilever, Uniform Distributed Load
$$ \delta = - { w x^2 \over 24EI } \left( 6L^2 - 4Lx + x^2 \right) $$

$$ \delta_{max} = {w L^4 \over 8EI} $$
@ x = L

$$ \theta = - {wx \over 6EI} \left( 3L^2 - 3Lx + x^2 \right) $$

$$ \theta_{max} = {w L^3 \over 6EI} $$
@ x = L

M_{max} = −wL^{2} / 2
@ x = 0

Cantilever, Triangular Distributed Load
$$ \delta = -{w_1 x^2 \over 120 LEI} \left( 10L^3 - 10 L^2 x + 5Lx^2 - x^3 \right) $$

$$ \delta_{max} = {w_1 L^4 \over 30EI} $$
@ x = L

$$ \theta = - {w_1 L \over 24LEI} \left( 4L^3 - 6 L^2 x + 4Lx^2 - x^3 \right) $$

$$ \theta_{max} = {w_1 L^3 \over 24EI} $$
@ x = L

V_{max} = +w_{1} L / 2
@ x = 0

M_{max} = −w_{1} L^{2} / 6
@ x = 0

Cantilever, End Moment
$$ \delta = - {M_0 x^2 \over 2EI} $$

$$ \delta_{max} = {M_0 L^2 \over 2EI} $$
@ x = L

$$ \theta = - {M_0 x \over EI} $$

$$ \theta_{max} = {M_0 L \over EI} $$
@ x = L

Simply Supported Beams
Fixed-Fixed Beams
Fixed-Fixed, Center Load
$$ \delta = -{Fx^2 \over 48EI} \left( 3L - 4x \right) $$
( 0 ≤ x ≤ L/2 )
$$ \delta_{max} = {F L^3 \over 192EI} $$
@ x = L/2

V_{1} = +F / 2
( 0 ≤ x ≤ L/2 )
V_{2} = −F / 2
( L/2 ≤ x ≤ L )

M = F (4x − L) / 8
( 0 ≤ x ≤ L/2 )

M_{1} = M_{3} = −FL / 8
@ x = 0 and x = L
M_{2} = +FL / 8
@ x = L/2

Fixed-Fixed, Uniform Distributed Load
$$ \delta = - { w x^2 \over 24EI } \left( L - x \right)^2 $$

$$ \delta_{max} = {w L^4 \over 384EI} $$
@ x = L/2

V_{1} = +wL / 2
@ x = 0
V_{2} = −wL / 2
@ x = L

M = w (6Lx − 6x^{2} − L^{2} ) / 12

M_{1} = M_{3} = −wL^{2} / 12
@ x = 0 and x = L
M_{2} = −wL^{2} / 24
@ x = L/2

Check out our beam calculator based on the methodology described here.

Calculates stresses and deflections in straight beams
Builds shear and moment diagrams
Can specify any configuration of constraints, concentrated forces, and distributed forces
References
Budynas-Nisbett, "Shigley's Mechanical Engineering Design," 8th Ed.
Gere, James M., "Mechanics of Materials," 6th Ed.
Lindeburg, Michael R., "Mechanical Engineering Reference Manual for the PE Exam," 13th Ed.