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Bahir Dar Institute of Technology
Department Of Civil Engineering
(Structural Engineering Stream)
Matrix Methods of Structural Analysis
4. Direct Stiffness Method
Seyfe N.
The Direct Stiffness Method
• The Direct Stiffness Method is a matrix-based numerical technique used in structural analysis to
calculate the response (displacements, forces, and stresses) of structures under loads.
• It is a fundamental approach in the Finite Element Method (FEM) and is particularly well-suited for
analyzing indeterminate structures such as trusses, beams, and frames.
• Structure divided into discrete elements
• Stiffness of elements combined into a global system
• Solving for nodal displacements, reactions, and internal forces
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The Direct Stiffness Method
Steps in the Direct Stiffness Method
1. Discretize the structure (elements and nodes)
2. Develop element stiffness matrices
3. Transform to global coordinate system
4. Assemble global stiffness matrix
5. Apply boundary conditions
6. Solve for displacements
7. Compute element forces
The Direct Stiffness Method
• In DSM the structure stiffness matrix is assembled directly from stiffness
matrix of the elements relating actions and displacements of both ends of the
element written in a global (structure) coordinate.
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Truss Element
Global Coordinate Displacements
Global Coordinate
Global Coordinate End Actions
Truss Element
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Truss Element
Truss Element
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Truss Element
Truss Element
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Truss Element
Truss Element
• Multiply with force
transformation matrix
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Truss Element
Assembling global stiffness matrix
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Assembling global stiffness matrix
For Member 1 For Member 2
Assembling global stiffness matrix
For Member 3
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Assembling global stiffness matrix
1 2
Imposing boundary conditions
1 2
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Member θ
1 90
2 0
3 90
4 0
5 50.19
6 50.19
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Example – Beam Analysis
Analyze the continuous beam shown in Figure below. Assume that the
supports are unyielding. Also assume that EI is constant for all members.
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Example – Beam Analysis
• The numbering of joints and members
• The possible global degrees of freedom (Numbers are put for the unconstrained (free) degrees
of freedom first and then that for constrained displacements.
Example – Beam Analysis
• The given continuous beam is divided into three beam elements Two degrees of freedom
(one translation and one rotation) are considered at each end of the member.
• In the given problem some displacements are zero, i.e., u3=u4=u5=u6=u7=u8=0 from
support conditions
• In the case of beams, it is not required to transform member stiffness matrix from local co-
ordinate system to global co-ordinate system, as the two co-ordinate system are parallel to
each other.
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Example – Beam Analysis
Example – Beam Analysis
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Example – Beam Analysis
• The stiffness matrix is symmetrical.
• The stiffness matrix is partitioned to separate the actions associated with two
ends of the member.
Example – Beam Analysis
• Member 1: L = 4m, node points 1- Member 2: L=4m, node points 2-3.
2.
Member 3:L=4m, node points 3-4.
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Example – Beam Analysis
• The assembled global stiffness matrix of the continuous beam is of the order 8x8.
Example – Beam Analysis
• Replace the given members loads by equivalent joint loads
Thus the global load vector corresponding to unconstrained degree of freedom is,
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Example – Beam Analysis
Writing the load displacement relation for the entire continuous beam
Example – Beam Analysis
• Thus solving for unknowns (u1 ,u2) and , u3=u4=u5=u6=u7=u8=0
The unknown joint loads are given by,
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Example – Beam Analysis
The actual reactions at the supports are calculated as,
Example – Beam Analysis
• Member end actions
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Example – Beam Analysis
Example – Beam Analysis
5
-5
5
-5
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Example – Beam Analysis
Beams
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Support
displacements
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