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/m^{3}
Calculate the max. deflection, max. shear force, max. bending moment, midspan
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 SUMMARY 
Parameter 
Symbol 
Value 
Unit 
Timber width 
b 
100 
mm 
Timber height 
H 
200 
mm 
Timber length 
L 
3000 
mm 
Distance of x (Midspan) 
x 
1500 
mm 
Elastic modulus of timber 
E 
8 
GPa 
Type of Beam Design 
Simply supported beam
with multiple point loads 

INPUT PARAMETERS 
Parameter 
Symbol 
Value 
Unit 
Height 
H 
200 
mm 
Width 
B 
100 
Length 
L 
3000 
Density 
p 
600 
kg/m^{3} 
OUTPUT PARAMETERS 
Parameter 
Symbol 
Value 
Unit 
Cross section area 
A 
20000 
mm^2 
Mass 
M 
36 
kg 
Second moment of area 
I_{xx} 
66666668 
mm^4 
Second moment of area 
I_{yy} 
16666667 
Section modulus 
S_{xx} 
666666.688 
mm^3 
Section modulus 
S_{yy} 
333333.344 
Radius of gyration 
r_{x} 
57.735 
mm 
Radius of gyration 
r_{y} 
28.868 
CoG distance in x direction 
x_{cog} 
50 
mm 
CoG distance in y direction 
y_{cog} 
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 ** 
P_{1} 
10 
0.5 
Load 2 ** 
P_{2} 
5 
1.5 
Load 3 ** 
P_{3} 
10 
2.5 
Load 4 ** 
P_{4} 
0 
0 
Load 5 ** 
P_{5} 
0 
0 
CONCENTRATED MOMENTS 
Parameter 
Symbol 
Magnitude 
Distance 
N*m 
m 
Moment 1 ** 
M_{1} 
0 
0 
Moment 2 ** 
M_{2} 
0 
0 
Moment 3 ** 
M_{3} 
0 
0 
Moment 4 ** 
M_{4} 
0 
0 
Moment 5 ** 
M_{5} 
0 
0 
DISTRIBUTED LOADS 
Parameterr 
Symbol 
Magnitude 
Distance 
N/m 
m 
wa 
wb 
a 
b 
Distributed Load 1 ** 
w_{1} 
117.7 
117.7 
0 
3 
Distributed Load 2 ** 
w_{2} 
0 
0 
0 
0 
Distributed Load 3 ** 
w_{3} 
0 
0 
0 
0 
Distributed Load 4 ** 
w_{4} 
0 
0 
0 
0 
Distributed Load 5 ** 
w_{5} 
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 

DISTRIBUTED LOADS 
No. 
Start Location 
Magnitude 
End Location 
Magnitude 
1 
0 m 
117.7 N/m 
3 m 
117.7 N/m 

RESULTS 

Parameter 
Symbol 
Value 
Unit 
Reaction Force 1 
R_{1} 
12676.5

N

Reaction Force 2 
R_{2} 
12676.5

Transverse Shear Force
@ distance x 
V_{x} 
2500.0

Maximum Transverse Shear Force 
V_{max} 
12676.5 
Moment @ distance x 
M_{x} 
8882.4 
N*m 
Maximum Moment 
M_{max} 
8882.4

Slope 1 
θ_{1} 
0.988

degree

Slope 2 
θ_{2} 
0.988

Slope
@ distance x 
θ_{x} 
0.000 
Maximum Slope 
θ_{max} 
0.988 
Deflection @ distance x 
y_{x} 
15.662 
mm 
Maximum Deflection 
y_{max} 
15.662 
Bending Stress @ distance x 
σ_{x} 
6.7 
MPa 
Maximum Bending Stress 
σ_{max} 
6.7 
Summary
Max. deflection, max. shear force, max. bending moment, midspan deflection/slope and end reaction forces of the timber rectangular beam have been calculated with the
usage of the following calculators.