Explanation Given information: Each A992 steel member has a cross-sectional area of A = 1.5 in . 2 . Method adopted: Castigliano’s theorem to determine displacement and Method of Joints to determine the force in the member. Assumptions: Young’s modulus or modulus of elasticity of A992 steel is E = 29 ( 10 3 ) ksi . Explanation: Castigliano’s theorem for the entire truss is as follows: Δ = ∑ N ( ∂ N ∂ P ) L A E (1) Here, Δ is displacement of the truss joint, P an external force of variable magnitude applied to the truss joint in the direction of Δ , N is internal axial force in a member caused by both force P and the actual loads of the truss, L is length of a member, A is cross-sectional area of a member, and E is modulus of elasticity of a member. Internal axial force ( N ) in the member calculation: Apply a vertical external force P in the direction of displacement to be determined. Determine the reactions: Entire truss: Show the free-body diagram of the entire truss as on Figure 1. Moment about point A : Determine the vertical reaction at point C by taking moment about the point A . ∑ M A = 0 C y ( 16 ) − 2 ( 16 ) − P ( 8 ) = 0 (2) Along the vertical direction: Determine the vertical reaction at point A by resolving the vertical component of forces. ∑ F y = 0 A y + C y − 2 − 2 − P = 0 (3) Along the horizontal direction: Determine the horizontal reaction at point A by resolving the horizontal component of forces. ∑ F x = 0 A x = 0 Show the calculation of values as follows: Solve Equation (2). C y ( 16 ) − 32 − 8 P = 0 C y = 2 + 0.5 P Substitute ( 2 + 0.5 P ) for C y in Equation (3). A y + 2 + 0.5 P − 4 − P = 0 A y = 2 + 0.5 P Joint C : Show the free-body diagram of the joint C as in Figure 2. Along the vertical direction: Determine the internal force in the member CD by resolving the vertical component of forces. ∑ F y = 0 C y − N C D = 0 (4) Along the horizontal direction: Determine the internal force in the member BC by resolving the horizontal component of forces. ∑ F x = 0 N B C = 0 Show the calculation of internal force in the members as follows: Substitute ( 2 + 0.5 P ) for C y in Equation (4). 3 + 0.5 P − N C D = 0 N C D = ( 2 + 0.5 P ) ( C ) Joint D : Show the free-body diagram of the joint D as in Figure 3. Determine the value of θ : tan θ = Opposite side Adjacent side (5) Along the vertical direction: Determine the internal force in the member BD by resolving the vertical component of forces. ∑ F y = 0 N C D − N B D sin θ − 2 = 0 (6) Along the horizontal direction: Determine the internal force in the member ED by resolving the horizontal component of forces. ∑ F x = 0 N E D − N B D cos θ = 0 (7) Show the calculation of internal force in the member as follows: Substitute 6 ft for opposite side and 8 ft for adjacent side in Equation (5). tan θ = 6 8 θ = 36.87 ° Substitute ( 2 + 0.5 P ) for N C D and 36.87 ° for θ in Equation (6). 2 + 0.5 P − N B D sin 36.87 ° − 2 = 0 N B D = 0.5 P sin 36.87 ° N B D = ( 0.833 P ) ( T ) Substitute 0.833 P for N B D and 36.87 ° for θ in Equation (7). N E D − ( 0.833 P ) cos 36.87 ° = 0 N E D = ( 0.667 P ) ( C ) Joint B : Show the free-body diagram of the joint B as in Figure 4. Along the vertical direction: Determine the internalforce in the member BE by resolving the vertical component of forces. ∑ F y = 0 N B D sin θ − N B E = 0 (8) Along the horizontal direction: Determine the internal force in the member AB by resolving the horizontal component of forces

9th Edition

ISBN: 9780133254426

Author: Russell C. Hibbeler

Publisher: Prentice Hall

See similar textbooks

Concept explainers

Axial Load

When a bar of the uniform cross-section is acted by a load, such that the load acts along the longitudinal axis of the bar, such kinds of loadings are known as axial loadings. In general terms, a load is an external force acting on a component. An axial …

Videos

Question

Chapter 14.9, Problem 14.126P

To determine

The vertical displacement of joint E.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 14.9, Problem 14.125P

Chapter 14.9, Problem 14.127P

Students have asked these similar questions

please show all steps

Determine the vertical displacement of joint B. Each A992 steel member has a cross-sectional area of 1.5 in2.

Determine the vertical displacement of joint B. Each A992 steel member has a cross-sectional area of 1.5 i

Chapter 14 Solutions

Mechanics of Materials

Ch. 14.2 - A material is subjected to a general state of...Ch. 14.2 - The strain-energy density for plane stress must be...Ch. 14.2 - 14–3. Determine the strain energy in the stepped...Ch. 14.2 - Prob. 14.4P Ch. 14.2 - Using bolts of the same material and...Ch. 14.2 - If P = 60 kN, determine the total strain energy...Ch. 14.2 - Determine the maximum force P and the...Ch. 14.2 - *14–8. Determine the torsional strain energy in...Ch. 14.2 - Determine the torsional strain energy in the A-36...Ch. 14.2 - The shaft assembly is fixed at C. The hollow...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.

The vertical displacement of joint E .

Solution Summary: The author explains Castigliano's theorem for the entire truss and Method of Joints to determine the force in the member.

Concept explainers

Videos

The vertical displacement of joint E.

Want to see the full answer?

Chapter 14 Solutions