What are the unknowns in the force method and the displacement method?

Redundant forces; joint displacements

Joint displacements; redundant forces

Which pairing is correct?

Force method: compatibility, flexibility matrix; displacement method: equilibrium, stiffness matrix

Force method: equilibrium, stiffness matrix

Displacement method: compatibility, flexibility matrix

Both: compatibility, stiffness matrix

2072 Kartik · 6 marks

Differences between force and displacement methods

Short answer

Force method uses redundant forces as unknowns and satisfies compatibility, suited for structures with few redundants. Displacement method uses joint displacements as unknowns and satisfies equilibrium, is systematic and preferred for computer analysis of frames with many joints.

List the differences between force and displacement methods. Draw a neat sketch of a system and explain.

The force (flexibility) method takes the redundant forces as unknowns and satisfies compatibility of displacements. The displacement (stiffness) method takes the joint displacements as unknowns and satisfies equilibrium of the joints.

Point	Force method	Displacement method
Unknowns	Redundant forces or moments ( $D_s$ )	Joint displacements ( $D_k$ )
Governing condition	Compatibility of displacements	Equilibrium of joints
Matrix used	Flexibility matrix $[F]$	Stiffness matrix $[K]$
Equations	$[F]\{X\} = -\{\Delta_0\}$	$[K]\{\Delta\} = \{P\}$
Primary structure	Statically determinate (redundants removed)	Kinematically determinate (all joints fixed)
Best for	Structures with few redundants (low $D_s$ ), trusses, arches	Structures with few joint displacements (low $D_k$ ), multi-storey frames, continuous beams
Suitable for computer	Less systematic, since the primary structure is not unique	Very systematic, so it is the basis of computer programs
Effect of settlement/temperature	Enters through the load vector $\Delta_0$	Enters through fixed-end forces

Illustration: portal frame with fixed bases

        B ______ C
        |        |
        |        |
       A///    ///D     (both fixed)

For the portal frame with fixed bases, $D_s = 3$ and $D_k = 3$ (rotations of B and C, one sway) if axial deformation is neglected. If one base is replaced by a hinge, $D_s = 2$ , but $D_k$ becomes 4 (the rotation at the hinge is added), or stays 3 if the modified stiffness $3EI/L$ is used for that column. A two-span continuous beam with hinged ends on rigid supports has $D_s = 1$ (the middle reaction) and $D_k = 3$ (rotations at the three supports), or $D_k = 1$ (rotation at the middle support only) if the modified stiffness is used for both end spans.

Force method steps: remove the redundants, find $M_0$ and $m_i$ , find $f_{ij}$ and $\Delta_{i0}$ , solve $[F]\{X\} = -\{\Delta_0\}$ , then superpose.

Displacement method steps: fix all joints, apply fictitious restraints, find the fixed-end forces, find the stiffness coefficients, solve $[K]\{\Delta\} = \{P\}$ for the joint displacements, then find the member end moments.

Choose the method that gives fewer unknowns: a frame with many redundant members but few joints suits the displacement method, whereas an arch or a truss with one or two redundants suits the force method.

✓ Answer checked by Claude Opus 5.5 (high effort)

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Differences between force and displacement methods

Illustration: portal frame with fixed bases

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