Q5. Figure 5 Figure 5 shows a cantilever beam with a hollow circular cross-section, loaded at point B. (a) In your own words, define Castigliano's theorem and modified Castigliano theorum and explain when you might need to use a 'dummy load'. (b) Using the modified form of Castigliano’s theorum, find an expression for the deflection of the beam at Point A. Clearly show all your working out. Include a sketch of the beam. (c) If the outer diameter of the beam is 30mm; wall thickness = 2 mm; Young’s Modulus, E = 205 GPa; total beam length, L = 1.5 m and load P = 600 N, calculate the following: (i) the second moment of area, I, of the beam (ii) the flexural rigidity of the beam (iii) the vertical deflection of the beam at point A.
Added by Chegg H.
Close
Step 1
Step 1: Understand the statement of Castigliano's Theorem, which states that the first partial derivative of strain energy (U) with respect to any generalized displacement is equal to the corresponding generalized force. Show more…
Show all steps
Your feedback will help us improve your experience
Carson Merrill and 55 other Physics 101 Mechanics educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
An overhanging beam $A B C$ has a guided support at $A,$ a rectangular cross section, and supports an upward uniform load $q=P / L$ over $A B$ and a downward concentrated load $P$ at the free end $C$ (see figure). The span length from $A$ to $B$ is $L$, and the length of the overhang is $L / 2 .$ The cross section has a width of $b$ and a height $h .$ Point $D$ is located midway between the supports at a distance $d$ from the top face of the beam. Knowing that the maximum tensile stress (principal stress at point $D$ is $\sigma_{1}=38$ MPa, determine the magnitude of the load $P$. Data for the beam are $L=1.75 \mathrm{m}$, $b=50 \mathrm{mm}, h=220 \mathrm{mm},$ and $d=55 \mathrm{mm}$.
Adriano C.
a) Determine the Euler buckling load for a 5m long, pin constrained strut, with a constant wall thickness cross-section profile, as shown in Fig. 2.1. The strut is made from aluminium extrusion with a Young's Modulus of 70GN/m". Confirm that is valid to use the Euler buckling analysis to determine the critical buckling load for this strut. b) For a rectangular cross-sectional profile cantilever beam, of length L, loaded at its end with load F, as shown in Fig. 2.2, determine an expression for the strain energy, U, due to shear force and bending moment, respectively. Using Castigliano's first theorem, determine a general equation for the defection under the load F.
Sri K.
Madhur L.
Recommended Textbooks
University Physics with Modern Physics
Physics: Principles with Applications
Fundamentals of Physics
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD