STRUCTURAL ANALYSIS
Consistent
Deformation
Method
Force Method · Complete Study Guide
CONTENTS
Beams 4
Continuous · propped cantilever
Frames 5
Portal · cantilever · inclined members
Trusses 2
Internally & externally indeterminate
Problems that frequently appear on structural
analysis exams — fully solved, step by step.
11 Fully Worked Problems · SFD & BMD in Every Solution
,Consistent Deformation Method (Force Method)
The Consistent Deformation (Force) Method enforces geometric compatibility by
treating redundant forces as the primary unknowns to systematically solve statically
indeterminate structures.
Basic Concept
1. Determine Indeterminacy: Calculate the degree of static indeterminacy n (num-
ber of redundant forces).
2. Select Primary Structure: Remove n redundant forces (reactions or internal
moments) to obtain a stable, statically determinate primary structure.
3. Calculate Load Displacements: Compute the deflection or rotation (∆L ) at
each redundant location due to the actual applied loads.
4. Determine Flexibility Coefficients: Apply a unit force or moment at each
redundant coordinate individually to obtain the flexibility matrix coefficients (fij
or δ).
5. Apply Compatibility: Write the compatibility equations requiring the total dis-
placement at each redundant coordinate to equal the actual support condition,
usually zero:
∆L + [f ] · {R} = {0}
For a single redundant: ∆load + δ · R = 0
6. Solve and Superimpose: Solve the linear system for the unknown redundant
forces {R}, then use statics to find all remaining reactions. Superimpose the ef-
fects to obtain final internal forces and construct shear force and bending moment
diagrams.
, Load w
(a) Original Structure
A B
(b) Primary Structure
under Load ∆B0 (down)
RB (Redundant)
δBB RB (up)
(c) Primary Structure
under Redundant
Compatibility Equation:
∆B0 + δBB RB = 0
, Problem Statement
Analyze the beam shown in Figure 1 by using consistent deformation method (Force
Method). Draw shear force and bending moment diagram for the beam. Consider any
two reactions as the redundant. Assume constant flexural rigidity EI.
w = 3 kips/ft
A B C D
10 ft 12 ft 10 ft
Figure 1: Continuous beam subjected to partial uniform load.
Degree of Indeterminacy & Primary Structure
Neglecting axial deformations, the reactions are Ay , By , Cy , Dy .
• Unknown vertical reactions = 4
• Equilibrium equations = 2 ( Fy = 0,
P P
M = 0)
• Degree of static indeterminacy = 4 − 2 = 2
Select reactions at B and C (RB and RC ) as redundants. The primary structure is a
simply supported beam pinned at A and on a roller at D.
Compatibility Equations:
∆B + RB δBB + RC δBC = 0 (1)
∆C + RB δCB + RC δCC = 0 (2)
Consistent
Deformation
Method
Force Method · Complete Study Guide
CONTENTS
Beams 4
Continuous · propped cantilever
Frames 5
Portal · cantilever · inclined members
Trusses 2
Internally & externally indeterminate
Problems that frequently appear on structural
analysis exams — fully solved, step by step.
11 Fully Worked Problems · SFD & BMD in Every Solution
,Consistent Deformation Method (Force Method)
The Consistent Deformation (Force) Method enforces geometric compatibility by
treating redundant forces as the primary unknowns to systematically solve statically
indeterminate structures.
Basic Concept
1. Determine Indeterminacy: Calculate the degree of static indeterminacy n (num-
ber of redundant forces).
2. Select Primary Structure: Remove n redundant forces (reactions or internal
moments) to obtain a stable, statically determinate primary structure.
3. Calculate Load Displacements: Compute the deflection or rotation (∆L ) at
each redundant location due to the actual applied loads.
4. Determine Flexibility Coefficients: Apply a unit force or moment at each
redundant coordinate individually to obtain the flexibility matrix coefficients (fij
or δ).
5. Apply Compatibility: Write the compatibility equations requiring the total dis-
placement at each redundant coordinate to equal the actual support condition,
usually zero:
∆L + [f ] · {R} = {0}
For a single redundant: ∆load + δ · R = 0
6. Solve and Superimpose: Solve the linear system for the unknown redundant
forces {R}, then use statics to find all remaining reactions. Superimpose the ef-
fects to obtain final internal forces and construct shear force and bending moment
diagrams.
, Load w
(a) Original Structure
A B
(b) Primary Structure
under Load ∆B0 (down)
RB (Redundant)
δBB RB (up)
(c) Primary Structure
under Redundant
Compatibility Equation:
∆B0 + δBB RB = 0
, Problem Statement
Analyze the beam shown in Figure 1 by using consistent deformation method (Force
Method). Draw shear force and bending moment diagram for the beam. Consider any
two reactions as the redundant. Assume constant flexural rigidity EI.
w = 3 kips/ft
A B C D
10 ft 12 ft 10 ft
Figure 1: Continuous beam subjected to partial uniform load.
Degree of Indeterminacy & Primary Structure
Neglecting axial deformations, the reactions are Ay , By , Cy , Dy .
• Unknown vertical reactions = 4
• Equilibrium equations = 2 ( Fy = 0,
P P
M = 0)
• Degree of static indeterminacy = 4 − 2 = 2
Select reactions at B and C (RB and RC ) as redundants. The primary structure is a
simply supported beam pinned at A and on a roller at D.
Compatibility Equations:
∆B + RB δBB + RC δBC = 0 (1)
∆C + RB δCB + RC δCC = 0 (2)
