# SIMPLY SUPPORTED BEAM EXAMPLE

### Simply Supported Timber Beam Deflection Example

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. ### Solution:

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

 INPUT PROPERTIES Parameter Value 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) INPUT PARAMETERS Parameter Value Height [H] 200 mm Width [B] 100 Length [L] 3000 Density [p] 600 kg/m3

 OUTPUT PARAMETERS Parameter Value 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 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 Parameter 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 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.

 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
 No. Start Location Magnitude End Location Magnitude 1 0 m 117.7 N/m 3 m 117.7 N/m
RESULTS Parameter Value
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.