Truss Analysis Calculations
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1. Truss Geometry
The overall configuration of the 2-dimensional truss is shown in Figure 1. The specific node and member configurations are also summarized in Table 1
and Table 2 below.
The total span of the truss is 35 ft and overall height of the truss is 4 ft.
Figure 1: Truss global configuration
Node ID
X-Position (ft)
Y-Position (ft)
Fixity (if not free)
0
0
0
pin
1
5.83
1.33
--
2
5.83
0
--
3
11.7
2.67
--
4
11.7
0
--
5
17.5
4
--
6
17.5
0
--
7
23.3
2.67
--
8
23.3
0
--
9
29.2
1.33
--
10
29.2
0
--
11
35
0
roller
Table 1: Structure node geometry
Member ID
Start -> End Node
Length (ft)
0
0→1
5.984
1
1→3
5.984
2
3→5
5.984
3
5→7
5.984
4
7→9
5.984
5
9 → 11
5.984
6
0→2
5.833
7
2→4
5.833
8
4→6
5.833
9
6→8
5.833
10
8 → 10
5.833
11
10 → 11
5.833
12
1→2
1.333
13
1→4
5.984
14
3→4
2.667
Member ID
Start -> End Node
Length (ft)
15
3→6
6.414
16
5→6
4
17
6→7
6.414
18
7→8
2.667
19
8→9
5.984
20
9 → 10
1.333
Table 2: Structure member geometry
2. Applied Loading to Nodes
The loads applied to this truss structure are represented in Figure 2 and summarized in detail below in Table 3. Note that if a node is omitted from
Table 3, no loads have been applied to it.
Figure 2: Graphical representation of loads applied to the structure (arrow length not to scale)
Node ID
Fx (kips)
Fy (kips)
0
0
-3.4
1
0
-3.4
3
0
-3.4
5
0
-3.4
7
0
-3.4
9
0
-3.4
11
0
-3.4
Table 3: Applied loading to nodes
3. Truss Analysis Using the Direct Stiffness Method
With the truss geometry and loading defined above, the member forces and deflections are calculated using the direct stiffness method. It is assumed
that all members behave elastically and have sufficient strength at connections to transfer the required load to the member.
3.1 Member Stiffness Matrix
First, each member stiffness matrix is composed in the global coordinate system. For truss analysis, it is assumed that both ends of the member are
rotationally unconstrained so that each member will only be loaded axially. The member stiffness matrix in the global coordinate system will be a 4x4
matrix for a 2-dimensional truss. Each member will be defined as follows:
Member starting node
Member ending node
Member rotation angle from horizontal
Member direction
Figure 3: General member geometry definition
Having member properties:
L
Member length
A
Member cross-sectional area
E
Member material modulus of elasticity
In this analaysis, A and E have been set to the following values:
Member Type
Cross-sectional Area (in^2)
Elastic Modulus (ksi)
Top Chord
29.8
29000
Bottom Chord
29.8
29000
Web Members
11.2
29000
For simplicity in this general example, the following constants are calculated:
c=cosθ
s=sinθ
And a stiffness matrix is assembled for each member using the following equation:
c 2 cs -c 2 -cs
ki=
AE
L
cs
s 2 -cs -s 2
-c 2 -cs c 2 cs
-cs -s 2 cs
s2
For example, the stiffness matrix for member 0 is:
0.95
2
k 0 = 29.8 in * 29000 ksi
5.98ft
0.217
-0.95
-0.217
0.217 0.0497 -0.217 -0.0497
-0.95
-0.217
0.95
0.217
-0.217 -0.0497 0.217 0.0497
3.2 Global Structure Stiffness Matrix
All of the member stiffness matrices will be combined to form the global structure stiffness matrix, K, by grouping each nodal degree of freedom and
summing the attached member stiffness matrix elements. For this 2-dimensional truss with 12 nodes, the global stiffness matrix will be 24x24.
This operation yields the following structural stiffness matrix for the above defined truss:
285000 31400 -137000 -31400 -148000
31400
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-31400
-7170
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-137000 -31400 326000
51000
0
0
-51600
11800
0
0
0
0
0
0
0
0
0
0
-31400
-7170
51000
261000
0
-7170
11800
-2700
0
0
0
0
0
0
0
0
0
0
-148000
0
0
0
296000
0
0
0
-148000
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-244000
0
244000
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-137000 -31400
0
0
316000
43600
0
0
19100
0
0
0
0
0
0
0
0
-31400
-7170
0
0
43600
145000
0
0
0
-51600
11800
-148000
0
0
0
348000
-11800
0
0
11800
-2700
0
0
0
-122000 -11800
124000
0
7170
0
0
0
-137000 -31400
-244000 -31400
0
0
-137000 -31400
-137000 -31400 -41900
-122000 -31400
0
-7170
19100
-8750
0
0
0
0
0
0
0
0
-148000
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-7.28e275000
12
0
0
-137000
31400
0
0
0
0
95500
0
-81200
31400
-7170
0
0
0
0
0
0
0
0
0
0
-31400
-7170
0
0
-7.28e12
0
0
0
0
0
0
-41900
19100
-148000
0
0
0
380000
1.46e-41900
11
-19100 -148000
0
0
0
0
0
0
0
0
0
19100
-8750
0
0
0
-81200
1.46e11
98700
-8750
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-41900 -19100 316000
-43600
0
0
-137000
31400
0
0
0
0
0
0
0
0
0
0
31400
-7170
-19100
-8750
-43600
145000
0
-122000
31400
-7170
0
0
0
0
0
0
0
0
0
0
0
0
-148000
0
0
0
348000
11800
-51600
-11800
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-122000
11800
124000
-11800
-2700
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-137000
31400
-51600
-11800
326000
-51000
0
0
0
0
0
0
0
0
0
0
0
0
0
0
31400
-7170
-11800
-2700
-51000
261000
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-148000
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-244000
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-137000
31400
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
31400
-7170
-137000 31400
-19100
Structure Stiffness Matrix, K
3.3 Reduced Structure Stiffness Matrix
With the reactions at the structure supports being unknown, the structure stiffness matrix is reduced by removing the rows and columns which
correspond to the node support directions, resulting in the reduced structure stiffness matrix, KR :
326000
51000
0
51000
261000
0
0
0
0
296000
0
0
-244000
0
-137000 -31400
-31400
-137000 -31400
-51600
11800
0
0
0
0
0
0
0
0
0
0
0
0
-7170
11800
-2700
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-148000
0
0
0
0
0
0
0
0
0
0
0
0
0
244000
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
316000
43600
0
0
19100
0
0
0
0
0
0
0
0
-7170
0
0
43600
145000
0
-51600
11800
-148000
0
0
0
348000
-11800
11800
-2700
0
0
0
-122000 -11800
124000
-244000 -31400
-137000 -31400 -41900
-122000 -31400
-7170
19100
-8750
0
0
0
0
0
0
0
0
0
0
-148000
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-137000
31400
0
0
0
0
0
0
0
-81200
31400
-7170
0
0
0
0
0
0
0
0
0
0
-137000 -31400
0
0
-7.28e275000
12
0
0
0
0
-31400
0
0
-7.28e12
0
0
0
0
-41900
-7170
19100
-148000
0
0
0
-19100 -148000
0
0
0
0
0
1.46e11
-8750
0
0
0
0
0
0
0
0
0
19100
-8750
0
0
0
0
0
0
0
0
0
0
-41900 -19100 316000
-43600
0
0
-137000
31400
0
0
0
0
0
0
0
0
0
0
31400
-7170
-19100
-8750
-43600
145000
0
-122000
31400
-7170
0
0
0
0
0
0
0
0
0
0
0
0
-148000
0
0
0
348000
11800
-51600
-11800 -148000
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-122000
11800
124000
-11800
-2700
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-137000
31400
-51600
-11800
326000
-51000
0
0
0
0
0
0
0
0
0
0
0
0
0
0
31400
-7170
-11800
-2700
-51000
261000
0
-244000
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-148000
0
0
0
296000
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-244000
0
244000
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-137000
31400
-148000
0
3.4 Reduced structure force matrix
-81200
1.46e380000
-41900
11
0
Reduced Structure Stiffness Matrix
0
95500
-137000 31400
98700
-19100
Given the loads applied to the structure, as described in Table 3, the global force matrix, Q, is assembled to match the dimensional size of the reduced
structure stiffness matrix. Each node degree of freedom for the structure will match between the structure force and structure stiffness matrices. Since
the reactions at the constrained nodes are unknown until the analysis is completed, the node support direction forces are removed from the global
structure force matrix to create the reduced structure load matrix, QR :
0 -3.4 0 0 0 -3.4 0 0 0 -3.4 0 0 0 -3.4 0 0 0 -3.4 0 0 0
Structural Load Matrix (reduced)
3.5 Analysis for global displacements
The global nodal displacements are calculated by inverting the reduced stiffness matrix and multiplying it with the reduced structure force matrix.
The resulting displacement at each node along with known support displacements are given below:
Node ID
Δx (ft)
Δy (ft)
0
0
0
1
0.000875
-0.00501
2
0.000251
-0.00501
3
0.000883
-0.006
4
0.000502
-0.00601
5
0.000703
-0.00592
6
0.000703
-0.006
7
0.000523
-0.006
8
0.000904
-0.00601
9
0.000531
-0.00501
10
0.00115
-0.00501
11
0.00141
0
Table 4: Structure node displacements
3.6 Calculate member axial demands
Using the relative displacements of each member's start and end nodes along with a transformed stiffness matrix, the axial demand on a member, q, is
calculated as follows:
ΔSx
qi=
AE
L
-c -s c s
ΔSy
ΔEx
ΔEy
Where ΔSx is the displacement of the starting node in the x-direction for member i.
The member axial demands for the truss described above are displayed in Figure 4 and summarized in detail in Table 5 along with the member's
length. Tensile axial loads are represented as negative forces, and compression axial demands are represented as positive forces.
Figure 4: Structure member loading (kips)
Member ID
Length (ft)
Axial Demand (kips)
0
5.984
38.15
1
5.984
30.52
2
5.984
22.89
3
5.984
22.89
4
5.984
30.52
5
5.984
38.15
6
5.833
-37.19
7
5.833
-37.19
8
5.833
-29.75
9
5.833
-29.75
10
5.833
-37.19
11
5.833
-37.19
12
1.333
0
13
5.984
7.629
14
2.667
-1.7
15
6.414
8.178
16
4
-6.8
17
6.414
8.178
18
2.667
-1.7
19
5.984
7.629
20
1.333
0
Table 5: Structure member demand summary (+Compression/-Tension)
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