Trusses vs. Beams: Analysis and Design

Trusses vs. Beams: Analysis and Design 1
TRUSSES VS. BEAMS: ANALYSIS AND DESIGN
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Trusses vs. Beams: Analysis and Design 2
Table of Contents
1. Introduction.......................................................................................................................................3
2. Task 1.................................................................................................................................................3
2.1. Neutral axis................................................................................................................................4
2.2. Moment of inertia......................................................................................................................6
2.3. Maximum bending stress..........................................................................................................8
3. Task 2.................................................................................................................................................9
3.1. Support reactions.......................................................................................................................9
3.2. Moments...................................................................................................................................10
4. Task 3...............................................................................................................................................15
5. Task 4...............................................................................................................................................24
5.1. Suitable diameter for the truss elements................................................................................24
5.2. Most convenient design solution.............................................................................................25
6. Conclusion........................................................................................................................................27
References................................................................................................................................................29

Trusses vs. Beams: Analysis and Design 3
1. Introduction
The strength, stability and safety of a structure largely depends on the design of its
elements. The design entails the geometric properties and materials of the elements. These
elements have to be designed such that they are compatible and create an integrated structure.
The aim of this report is to analyze the structural behavior of simple structural elements – trusses
and beams, by performing different calculations. The report contains four tasks each with a
different question. The first task involves analyzing three different cross sections of a beam and
selecting the one that would be nest for design purposes. The second task involves analyzing a
simply supported overhang beam subjected to a uniformly distributed load by calculating its
external forces, drawing the shear force and bending moment diagrams, and calculating the
maximum normal stress that would be acting on the beam. The third task involves determining
external reactions and internal forces of in all members of a steel truss. The fourth task involves
determining a suitable diameter for the truss elements, and determining the most convenient
design solution between the beam in task 2 and the truss in task 3.
2. Task 1
Since the three cross sections have the same cross-sectional area and total height, and
they are subjected to the same bending load, the most suitable approach of determining the one
that would be the best design option for the beam is calculating the maximum bending stress in
each cross section. This will help in determining the capability of each cross section to withstand
the maximum bending moment of the beam.
The maximum bending stress of a beam is calculated using equation 1 below
σ max ¿ Mc
I ................................................... (1)

Trusses vs. Beams: Analysis and Design 4
Where σmax is the maximum bending moment; c is the vertical distance between the neutral axis
and the topmost or bottommost part of the beam (where the beam can experience either tension
or compression force), and I is the second moment of area (moment of inertia) of the beam.
In this case, the three cross sections have equal cross sectional area and total height hence the
maximum bending moment will also be the same in all the cross sections. However, the values of
c and I will be varied for each cross section because they have different geometries. Thus it is
important to calculate the values of c and I for each cross section.
2.1. Neutral axis
Calculating neutral axis, y, of the cross sections
i) First cross section
y= A 1 y 1
A 1
A1 = 40 cm x 80 cm = 3,200 cm2; y1 = 80 cm
2 =40 cm
y= A 1 y 1
A 1 = 3200 c m2 x 40 cm
3200 c m2 =40 cm
ii) Second cross section
y= A 1 y 1+ A 2 y 2
A 1+ A 2
Top segment: A1 = 32 cm x 48 cm = 1,536 cm2; y1 = 32cm + 48 cm
2 = 32 cm + 24 cm = 56 cm
Bottom segment: A2 = 52 cm x 32 cm = 1,664 cm2; y2 = 32cm
2 = 16 cm

Trusses vs. Beams: Analysis and Design 5
y= A 1 y 1+ A 2 y 2
A 1+ A 2 = ( 1536 c m2 x 56 cm ) +(1664 c m2 x 16 cm )
1536 c m2 +1664 c m2 = 86016+26624
3200 = 112640 c m3
3200 c m2 =35.2 cm
iii) Third cross section
y= A 1 y 1+ A 2 y 2+ A 3 y 3
A 1+ A 2+A 3
Top segment: A1 = 60 cm x 20 cm = 1,200 cm2; y1 = 20 cm + 40 cm + 20 cm
2 = 20 cm + 40 cm +
10 cm = 70 cm
Middle segment: A2 = 20 cm x 40 cm = 800 cm2; y2 = 20 cm + 40 cm
2 = 20 cm + 20 cm = 40 cm
Bottom segment: A3 = 60 cm x 20 cm = 1,200 cm2; y3 = 20 cm
2 =10 cm
y= A 1 y 1+ A 2 y 2+ A 3 y 3
A 1+ A 2+A 3 = ( 1200 c m2 x 70 cm ) + ( 800 c m2 x 40 cm ) +(1200 c m2 x 10 cm)
1200 c m2 +800 c m2 +1200 c m2
= 84000 c m3 +32000 c m3 +12000 c m3
3200 c m2 = 128000 c m3
3200 c m2 =40 cm
After calculating the values of neutral axis, these values are used to determine the values of c as
follows:
First cross section:
c = total height – y = 80 cm – 40 cm = 40 cm. This can either be at the topmost or bottommost
part of the beam. It means that the beam will experience maximum bending stress either at the
topmost or bottommost part of the beam.
Second cross section:

Trusses vs. Beams: Analysis and Design 6
c = total height – y = 80 cm – 35.2 cm = 44.8 cm. This is at the topmost part of the beam. It
means that the beam will experience maximum bending stress either at the topmost part of the
beam.
Third cross section:
c = total height – y = 80 cm – 40 cm = 40 cm. This can either be at the topmost or bottommost
part of the beam. It means that the beam will experience maximum bending stress either at the
topmost or bottommost part of the beam.
2.2. Moment of inertia
Calculating the moment of inertia, I, of the three cross sections
The moment of inertia of the three cross sections is calculated using parallel axis theorem (1).
The formula used for the calculations is provided in equation 2 below (2).
Itotal = ∑ (Ii+ Aihi2 )........................................... (2)
Where Ii = moment of inertia of individual segment; Ai = area of individual segment; and hi =
vertical distance between neutral axis and centroid of the segment.
And Ii is calculated using equation 3 below
I = b d3
12 ................................................. (3)
Where b = breadth, and d = depth of the beam.
i) First cross section
b = 40 cm, d = 80 cm, A1 = 3200 cm2, h1 = 40 cm – 40 cm = 0 cm