CSM $325$ Structural and Soil Mechanics in Construction

Homework $10$: Bending stresses in an LVL beam

Given: An engineered wooden beam of Laminated Veneer Lumber $($LVL$)$ with $h=11\mathrm{.}25$ inches and $b=1\mathrm{.}75$ inches is to span a distance of $18$ feet with a horizontal spacing of $2$ feet. The beam is to carry a distributed floor load of $100l\frac{b}{f}{t}^2.$ Assume the end constraints on the beam serve to act as simply supported.

The LVL design properties from table below:

Maximum allowable stress from moment $=F_b=2900$ psi

Maximum allowable stress from shear $=F_v=285$ psi

The modulus of elasticity $E=2{10}^6$

Find:

a$)$ The actual working stresses $($both bending and shear$)$ under the described loading, and compare the working stresses with the allowable stresses to show that the member is adequate.

b$)$ Calculate the deflection under the beam. $=\frac{5w{L}^3}{384EI}$ where $w$ is the distributed load, $L$ is the span length, $E$ is the modulus of elasticity, and I is the moment of inertia. Note: feet do not equal inches.

Steps:

Find the distributed load $w$ acting on the beam $($hint: it is not $100l\frac{b}{f}{t}^2)$

Find the reactions for the beam

Draw the shear diagram for the beam

Draw the bending moment diagram for the beam

Determine the maximum shear force $V_{\text{m}\text{a}\text{x}}$ that must be resisted.

Determine the maximum moment $M_{\text{m}\text{a}\text{x}}$ that must be resisted.

Determine the moment of inertia I for the beam

Determine the maximum extreme fiber bending stress due to the maximum bending moment $f_b$ actual $=M\frac{c}{\text{I}}$

Compare $f_b$ actual with $F_b$ allowable for the beam. Is the beam adequate in Moment capacity?

Determine the factor of safety relative to the allowable stresses $=F{S}_{\text{b}\text{e}\text{n}\text{d}\text{i}\text{n}\text{g}}=\frac{F_{b\text{a}\text{l}\text{l}\text{o}\text{w}\text{a}\text{b}\text{l}\text{e}}}{f_{b\text{a}\text{c}\text{t}\text{u}\text{a}\text{l}}}$

Determine the maximum shear stress in the beam $f_{v\text{a}\text{c}\text{t}\text{u}\text{a}\text{l}}=\frac{V_{\text{m}\text{a}\text{x}}}{A}$ where $A$ is the cross$-$sectional area of the beam

Compare $f_v$ actual with $F_{\text{v}\text{a}\text{l}\text{l}\text{o}\text{w}\text{a}\text{b}\text{l}\text{e}}$ for the beam. Is the beam adequate in Shear?

Determine the factor of safety relative to the allowable stresses $=F{S}_{\text{s}\text{h}\text{e}\text{a}\text{r}}=\frac{F_{\text{v}\text{a}\text{l}\text{l}\text{o}\text{w}\text{a}\text{b}\text{l}\text{e}}}{f_v}$ actual

Based on your calculations, does shear or moment capacity govern the response of this beam? Or$,$ why is it customary to determine the moment capacity before checking the shear capacity?

Determine the theoretical deflection of this beam under the prescribed loading.