Sample Problem : Timber Beam Deflection Calculations


A timber beam AB of span 3 m, width 200 mm and height 100 mm is to support three concentrated loads shown in the figure. Modulus of elasticity of selected class of timber is 8 GPa and the density of the timber is 600 kg/m3

Calculate the max. deflection, max. shear force, max. bending moment, mid-span deflection/slope and end reaction forces of the timber rectangular beam for the following loading conditions.

Timber Beam Design For Strength Example
Solution:


Step 1 : Write down input parameters (including material properties) which are defined in the sample example.



INPUT PROPERTIES SUMMARY
Parameter Symbol Value Unit
Timber width b 100 mm
Timber height H 200 mm
Timber length L 3000 mm
Distance of x (Mid-span) x 1500 mm
Elastic modulus of timber E 8 GPa
Type of Beam Design Simply supported beam
with multiple point loads


Step 2 : Go to " Sectional Properties Calculator of Solid Rectangular Bar"  page to calculate second moment of area around x axis(Ixx)


Sectional properties of solid rectangle
INPUT PARAMETERS
Parameter Symbol Value Unit
Height H 200 mm
Width B 100
Length L 3000
Density p 600 kg/m3


OUTPUT PARAMETERS
Parameter Symbol Value Unit
Cross section area A 20000 mm^2
Mass M 36 kg
Second moment of area Ixx 66666668 mm^4
Second moment of area Iyy 16666667
Section modulus Sxx 666666.688 mm^3
Section modulus Syy 333333.344
Radius of gyration rx 57.735 mm
Radius of gyration ry 28.868
CoG distance in x direction xcog 50 mm
CoG distance in y direction ycog 100

Step 3 : Go to "Simply Supported Beam Stress and Deflection Calculator"  page to calculate maximum shear force, bending moment and deflections on the timber. Enter three point loads given in the figure and one distributed load (due to the timber beam's own weight). Distributed load is equal to (M*g)/L = 36 * 9.81 / 3 = 117.7 N/m .

There is no moment acting on the timber beam so set moment values to 0.


INPUT PARAMETERS
POINT LOADS
Simply Supported Beam Point Loading
Parameter Symbol Magnitude Distance
kN m
Load 1 ** P1 10 0.5
Load 2 ** P2 5 1.5
Load 3 ** P3 10 2.5
Load 4 ** P4 0 0
Load 5 ** P5 0 0
CONCENTRATED MOMENTS
Parameter Symbol Magnitude Distance
N*m m
Moment 1 ** M1 0 0
Moment 2 ** M2 0 0
Moment 3 ** M3 0 0
Moment 4 ** M4 0 0
Moment 5 ** M5 0 0
DISTRIBUTED LOADS
Parameterr Symbol Magnitude Distance
N/m m
wa wb a b
Distributed Load 1 ** w1 117.7 117.7 0 3
Distributed Load 2 ** w2 0 0 0 0
Distributed Load 3 ** w3 0 0 0 0
Distributed Load 4 ** w4 0 0 0 0
Distributed Load 5 ** w5 0 0 0 0
STRUCTURAL BEAM PROPERTIESS
Parameter Symbol Value Unit
Beam Length L 3 m
Distance x x 1.5
Modulus of Elasticity E 8 GPa
Distance from neutral axis to extreme fibers c 50 mm
Second moment of area I 66666668 mm^4


Step 4 : Calculation results of step 3 are as follows.


INPUT LOADING TO SIMPLY SUPPORTED BEAM
POINT LOADS
No. Location Magnitude
1 0.5 m 10 kN
2 1.5 m 5 kN
3 2.5 m 10 kN
CONCENTRATED MOMENTS
No. Location Magnitude
DISTRIBUTED LOADS
No. Start Location Magnitude End Location Magnitude
1 0 m 117.7 N/m 3 m 117.7 N/m
RESULTS
Simply Supported Beam Deflected
Parameter Symbol Value Unit
Reaction Force 1  R1 12676.5 N
Reaction Force 2 R2 12676.5
Transverse Shear Force
@ distance x
Vx 2500.0
Maximum Transverse
 Shear Force
Vmax 12676.5
Moment @ distance x Mx 8882.4 N*m
Maximum Moment Mmax 8882.4
Slope 1  θ1 -0.988 degree
Slope 2 θ2 0.988
Slope
@ distance x
θx 0.000
Maximum Slope θmax -0.988
Deflection @ distance x yx -15.662 mm
Maximum Deflection ymax -15.662
Bending Stress @ distance x σx 6.7 MPa
Maximum Bending Stress σmax 6.7

Summary

Max. deflection, max. shear force, max. bending moment, mid-span deflection/slope and end reaction forces of the timber rectangular beam have been calculated with the usage of the following calculators.



Calculator Usage
Solid Rectangular Bar - Sectional Properties Calculator To calculate sectional properties of rectangular solid bar.
Simply Supported Beam Stress and Deflection Calculator To calculate forces, moments, stresses, deflections and slopes in a simply supported beam for multiple point loads, distributed loads and concentrated moments.