# Beam theory: Stiffness of combined loads

## Engineering Fundamentals

#### Introduction

Most beam theory examples use perfect loading conditions with often a single load. But what happens when the loads are not applied at the perfect location or when a combination of loads is applied? This page aims to give some feel for the change in stiffness by giving two examples; tension/compression combined with bending due to off-center loading and shear combined with bending.

### Off-center Tension/Compression

#### General energy equation

$U=\frac{C\cdot\Delta l^2}{2}=\frac{C\cdot\left(\frac{F}{C}\right)^2}{2}=\frac{F^2}{2C}$

$C_t=\frac{E\cdot w\cdot h}{l}$
$U_t=\frac{F_z^2}{2C_t}=\frac{F_z^2l}{2E\cdot w\cdot h}$

$\theta=\frac{M\cdot l}{E\cdot I_x}=\frac{12F_z\cdot u\cdot l}{E\cdot w\cdot h^3\ }$
$\delta_z=\theta\cdot u$
$C_b=\frac{F_z}{\delta_z}=\frac{E\cdot w\cdot h^3}{12\cdot u^2\cdot l}$
$U_b=\frac{F_z^2}{2C_b}=\frac{6F_z^2\cdot u^2\cdot l}{E\cdot w\cdot h^3}$

$U_c=U_t+U_b=\frac{F_z^2}{2}\left(\frac{l}{E\cdot w\cdot h}+\frac{12u^2\cdot l}{E\cdot w\cdot h^3}\right)$
$C_c=\frac{E\cdot w\cdot h}{l\left(1+12\left(\frac{u}{h}\right)^2\right)}$

$\frac{C_c}{C_t}=\frac{\frac{E\cdot w\cdot h}{l\left(1+12\left(\frac{u}{h}\right)^2\right)}}{\frac{E\cdot w\cdot h}{l}}=\frac{1}{1+12\left(\frac{u}{h}\right)^2}$

Note: when a load is applied to the outer fiber of the beam ($u=0.5h$) the stiffness $C_c$ of the beam will be 4 times as low as when the beam is loaded at its centerline:
$C_c=\frac{1}{4}\cdot\frac{E\cdot w\cdot h}{l}$

### Shear and Bending

#### Bending stiffness

$C_b=\frac{3EI_y}{l^3}=\frac{Ewh^3}{4l^3}$

#### Shear stiffness

$C_s=\frac{F_y}{\delta_y}=\frac{G\cdot A}{l}=\frac{\frac{Ewh}{2\left(1+\nu\right)}}{l}=\frac{0.385E\cdot w h}{l}$
Note: this assumes the shear stress has a uniform distribution over the cross section of the beam. Also at the location where the force is introduced!

#### Determinative stiffness depending on beam dimensions

In the figure shown below the bending stiffness $C_b$ and the shear stiffness $C_s$ are illustrated as a function of the ratio $\frac{l}{h}$ between the length $l$ and the height $h$ of the beam. When $\frac{l}{h}\approx0.8$ the bending and shear stiffness are equal. When $\frac{l}{h}<0.8$ the lower value of the shear stiffness is determinative, when $\frac{l}{h}>0.8$ the lower value of the bending stiffness is determinative.

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