p.3
Equilibrium Conditions
What is the equation for the first deformation due to external forces and redundants?
Δ1 = ΔF1 + δ11 Re1 + δ12 Re2 = 0.
What is the equation for virtual work used in the context of this problem?
V' B = 67,840 3 EI m and δ 11 = 512 3 EI m.
p.14
Indeterminate Structures
How is the analysis of an externally indeterminate truss conducted?
Using a similar method as for an indeterminate beam.
p.17
Base Structure Selection
What happens to Hinge e in the base structure selection?
It is changed to a roller.
p.11
Redundant Forces in Structures
What is the effect of the redundant force on the deflection at C?
It is represented as δ11 multiplied by the redundant force X1.
p.10
Method of Consistent Deformation
What is the first step in the Consistent Deformation Method?
Substitute displacements into the constraint equations.
What must the displacements of the base structure at points with redundant forces be consistent with?
Those of the original structure.
p.17
Equilibrium Conditions
What are the two constraint equations used in the truss analysis?
Δ1 = 0 (Horizontal displacement at e must = 0) and Δ2 = 0 (Gap in Member Cd must = 0).
p.2
Base Structure Selection
What is the first step in the example procedure?
Select the base structure.
p.9
Method of Consistent Deformation
What is the first step in the Consistent Deformation Method?
Select Base Structure: Simple beam with overhang.
p.18
Equilibrium Conditions
What is the second constraint equation formulated?
Δ2' + δ21 · X1 + δ22 · X2 = 0.
p.15
Truss Analysis Techniques
What method is used to find reactions at all supports in the truss example?
Method of consistent deformation.
p.15
Base Structure Selection
What is the base structure selected for the truss analysis?
A simple truss (removing Roller c).
p.12
Consistent Deformation Method
What is the consistent deformation method used for?
To calculate deformations in structures.
p.11
Indeterminate Structures
What are the two main outputs required from the analysis of the frame?
Reactions at all supports and the shear force diagram (SFD) and bending moment diagram (BMD).
p.14
Indeterminate Structures
What is the first step in analyzing an internally indeterminate truss?
Cut the redundant members.
p.8
Method of Consistent Deformation
What method is used to analyze the structure and find reactions at A and D?
Method of Consistent Deformation.
p.9
Equilibrium Conditions
What is the equation for vertical displacement at point A?
Δ1 = Δ1' + δ11 * Z1 + δ12 * Z2 = 0.
p.11
Indeterminate Structures
What method is used to find reactions at all supports in the frame example?
Method of consistent deformation.
p.11
Kinematic Constraints
What is the condition at support C in the cantilever frame?
End C is free, and the deflection at C must equal 0.
p.8
Method of Consistent Deformation
How do you find the moment (M_B) at member B?
Using equilibrium: ΣM(B) = 0; M_B + 13.3·4 = 0, then M_B = -53.3 kN·m.
p.4
Method of Consistent Deformation
What method is used to find reactions at all supports in Example 1?
Method of consistent deformation.
p.1
Method of Consistent Deformation
What is the relationship between the Structure of Interest and the Base Structure?
The Base Structure is adapted from the Structure of Interest.
p.3
Redundant Forces in Structures
What are the three separated effects considered for deformation?
External forces, Redundant Re 1, and Redundant Re 2.
p.7
Redundant Forces in Structures
What does the symbol Z represent in the context of redundant forces?
Redundant forces in structures.
p.10
Equilibrium Conditions
What equilibrium equations are used to calculate other reactions?
ΣF_X = 0; ΣM(C) = 0; ΣF_Y = 0.
p.2
Base Structure Selection
What must be similar in terms of forces in the base structure?
Redundant forces Re1 & Re2.
p.16
Equilibrium Conditions
What equilibrium conditions are considered for the whole structure?
ΣF_X = 0; ΣF_Y = 0; ΣM = 0.
What is the significance of the values Δ ’ 1 and Δ ’ 2 in the context of the analysis?
Δ ’ 1 = 96 and Δ ’ 2 = 27.2 represent the total deformations in the system.
p.3
Equilibrium Conditions
What is the equation for the second deformation due to external forces and redundants?
Δ2 = ΔF2 + δ21 Re1 + δ22 Re2 = 0.
p.8
Method of Consistent Deformation
What is the equation for substitute deformations in the constraint equation?
ΔA = Δ'A + Δ''A = 853.3/EI + 64Z/EI = 0.
p.5
Base Structure Selection
What is the first step in formulating the constraint equation for a cantilever beam?
Select 'Base Structure' as a cantilever beam.
p.18
Equilibrium Conditions
What is the first constraint equation formulated?
Δ1' + δ11 · X1 + δ12 · X2 = 0.
p.8
Method of Consistent Deformation
What are the constants in the structure analysis mentioned?
E (modulus of elasticity) and I (moment of inertia) are constants and similar in all members.
p.1
Method of Consistent Deformation
How is the Base Structure adapted from the Structure of Interest?
By taking out some redundant forces (reactions or internal forces).
p.14
Indeterminate Structures
What defines an internally indeterminate truss?
Redundant forces are axial forces.
p.3
Equilibrium Conditions
What is the purpose of using equilibrium equations in this context?
To obtain other unknown forces (R1, R2, & R3).
How do you express the second constraint in a structure with 3 degrees of freedom?
Δ2 = ΔF2 + δ21 Re1 + δ22 Re2 + δ23 Re3 = 0.
p.5
Indeterminate Structures
How is the deflection at B represented in the constraint equation?
v B = v ' B + δ 11 * Z 1 = 0.
p.5
Examples of Beam Analysis
What is the formula for calculating v ' B using the method of Virtual Work?
V' B = ∫ M * m / (E I) dx.
p.13
Truss Analysis Techniques
What is the significance of SFD and BMD in structural analysis?
They summarize internal forces in each section of the structure.
p.19
Indeterminate Structures
What is the meaning of the values -25.6 k and -10.64 k?
They represent the calculated deformations X 1 and X 2, indicating compression.
What does 'm' represent in the context of the example?
Moment from unit force at point C.
p.7
Base Structure Selection
What is the first step in formulating the constraint equation for a simple beam?
Select 'Base Structure' as a simple beam.
What is the constraint equation derived from the virtual work equations?
v A = 67,840 3 EI + 512 3 EI Z 1 = 0.
p.8
Method of Consistent Deformation
What are the components to be drawn after analyzing the structure using the Method of Consistent Deformation?
Shear Force Diagram (SFD) and Bending Moment Diagram (BMD).
p.6
Method of Consistent Deformation
What method is used to find reactions at Support A and the bending moment at B?
Method of consistent deformation.
p.13
Equilibrium Conditions
What are the reactions obtained from equilibrium?
H_A = -30.0 kN, V_A = 27.3 kN, M_A = 198.4 kN-m.
p.15
Truss Analysis Techniques
What is the first step in analyzing the truss?
Identify the redundant force and constrain equation.
p.14
Indeterminate Structures
What are the three types of indeterminate trusses?
Externally indeterminate, internally indeterminate, and externally & internally indeterminate.
p.10
Redundant Forces in Structures
What are the values of the redundant forces Z1 and Z2?
Z1 = 33.7 kN (Reaction at A) and Z2 = 54.0 kN (Reaction at D).
p.18
Indeterminate Structures
What are the two main effects considered in the constraint equations?
Horizontal displacement at point e and gap in Member Cd.
What is the general form of the constraint equation for N degrees of freedom?
Δ1 = ΔF1 + δ11 Re1 + δ12 Re2 + ... + δ1N ReN = 0.
What does δ 11 represent in the context of deflection?
Deflection at C under unit force representing X 1, δ 11 = 810.7 /EI.
p.16
Equilibrium Conditions
What does δ11 represent in the context of this example?
Vertical displacement at C under unit force representing X1.
p.19
Equilibrium Conditions
What does the equation 96 (L/EA) + 4 (L/EA) X 1 + 0.6 (L/EA) X 2 = 0 represent?
It represents the constraint equation for the system considering deformations.
p.15
Indeterminate Structures
What does δ11 represent in the context of the truss analysis?
Vertical displacement at C due to the redundant force X1.
What does 'M' represent in the context of the example?
Moment from external force.
p.17
Truss Analysis Techniques
What method is used to find the horizontal force at Support e in the truss example?
Method of consistent deformation.
p.10
Method of Consistent Deformation
What are the equations obtained after substituting displacements in Step 8?
Δ1 = -18,688 / EI + 384.0 Z1 / EI + 106.7 Z2 / EI = 0 and Δ2 = -24,320 / EI + 106.7 Z1 / EI + 384.0 Z2 / EI = 0.
What are the equations required for kinematics in Step 3 called?
Constrain equations (supplement equations).
p.14
Indeterminate Structures
What must be true about the gap between each cut in an internally indeterminate truss?
The gap (Δ1) must be zero.
p.18
Redundant Forces in Structures
What is the significance of δ11 and δ21 in the equations?
They represent the deformations due to the redundant forces X1 and X2.
p.9
Redundant Forces in Structures
What does δ11 represent in the context of the Consistent Deformation Method?
The displacement due to unit load representing Z1.
p.16
Superposition Principle
What is the formula for calculating axial force in each member?
Axial force = N + n · X1.
p.14
Indeterminate Structures
What defines an externally indeterminate truss?
Redundant forces are reactions.
p.10
Examples of Beam Analysis
What is the final step after obtaining reactions in Step 10?
Find internal forces and draw Shear Force Diagram (SFD) and Bending Moment Diagram (BMD).
p.5
Equilibrium Conditions
What is the condition for deflection at point B in the cantilever beam?
Deflection at B must equal 0 (v B = 0).
What is the significance of the integral in the virtual work equations?
It calculates the work done by bending moments.
p.18
Equilibrium Conditions
What is the relationship between deformation and axial force in the equations?
Deformation is proportional to the product of axial force, length, area, and modulus of elasticity.
p.13
Equilibrium Conditions
What are the equilibrium conditions for the whole structure?
ΣF_X = 0; ΣF_Y = 0; ΣM = 0.
p.19
Superposition Principle
How is the axial force in each member calculated?
Using the superposition principle: Axial force = N + n 1 * X 1 + n 2 * X 2.
p.14
Indeterminate Structures
What does the redundant force equal in an internally indeterminate truss?
Axial force in any cut member.
What is the equation for the first constraint in structures with 3 degrees of freedom?
Δ1 = ΔF1 + δ11 Re1 + δ12 Re2 + δ13 Re3 = 0.
What are the constraints for vertical displacements in the Consistent Deformation Method?
Δ1 = 0 at A and Δ2 = 0 at D.
How do you calculate displacements using the Method of Virtual Work?
Using the integrals of bending moments M, m1, and m2.
p.13
Equilibrium Conditions
How is X 1 calculated from the deformation equations?
Substituting deformations gives -42,720.0 /EI + 810.7 /EI * X 1 = 0, resulting in X 1 = 52.7 kN.
p.13
Truss Analysis Techniques
What are the internal forces summarized in the SFD?
SFD values are 27.3 kN, 30.0 kN, -52.7 kN, 101.6 kN, 138.9 kN.
p.18
Indeterminate Structures
What do N, n1, and n2 represent in the context of the problem?
They represent axial forces in the structure.
p.5
Examples of Beam Analysis
What is the bending moment M for the interval B to A?
M = -50 (x 1 + 4) - 15 X 1^2 / 2.
p.16
Equilibrium Conditions
What does Δ1' represent in the context of this example?
Vertical displacement at C under external forces.
p.13
Truss Analysis Techniques
What are the internal forces summarized in the BMD?
BMD values are +, +, +, -, -, +.
What do the variables X 1 and X 2 represent in the equations?
X 1 and X 2 represent the deformations due to axial forces in the members.
p.19
Indeterminate Structures
What is the role of the axial force induced by external forces?
It contributes to the overall axial force in the members of the structure.
p.1
Method of Consistent Deformation
What is the fundamental method used in the Method of Consistent Deformation?
Using existing knowledge of statics and kinematics of determinate structures.
