Deflection and Stiffness

A straight beam is supported on two cylinders such that the length between supports decreases as the beam is deflected by a force F. A larger force is required to deflect the beam the more it is deflected. How is this beam described in the context of spring behavior?

What is the formula for the total extension or contraction of a uniform bar in pure tension or compression, respectively?

Which equation represents the curvature of a beam subjected to a bending moment M, where EI is the flexural rigidity?

What are the three conditions that must be met to apply the principle of superposition for beam deflections?

What is the primary purpose of using singularity functions in the context of beam deflections?

What is the formula for the strain energy (U) in a beam subjected to a bending moment (M) that varies along its length?

What does Castigliano's theorem state regarding the displacement of a point of application of a force on an elastic system?

For a carbon steel bar 10 inches long with a 1-inch diameter (E = 30 Mpsi, G = 11.5 Mpsi) under a 100 lbf end load, what is the percent error introduced in the maximum deflection calculation if the effect of transverse shear is neglected?

What is a system called when it has more unknown support forces and/or moments than available static equilibrium equations?

According to the Euler column formula, what is the critical buckling load (Pcr) for a long column with pinned (rounded) ends?

What term is used to describe the ratio l/k for a column, where l is the length and k is the radius of gyration?

For designing steel columns, what is the recommended value for the end-condition constant C for a fixed-fixed column, considering practical limitations?

What is the primary characteristic of the J. B. Johnson formula used for intermediate-length columns?

A steel strut is 1 inch square and 4 inches long. It is loaded in compression with a force of 1000 lbf, offset from the centroidal axis by an eccentricity e = 0.10 inches. What is the maximum compressive stress in the strut?

In the study of shock and impact, a weight W falls from a height h onto a beam, causing a maximum deflection delta. By equating the loss of potential energy to the strain energy gained by the beam (modeled as a spring with rate k), which quadratic equation is derived for delta?

A round steel bar is subjected to a twisting moment T. The angular deflection in radians is given by theta = Tl / (GJ). How can this be rearranged to give the torsional spring rate, k?

For a simply supported beam of length l = 20 in with a uniform load w = 80 lbf/in, what is the bending moment M at any point x along the beam?

What is the strain-energy correction factor (C) for transverse shear in a beam with a rectangular cross-section?

How can Castigliano's theorem be used to find the deflection at a point on a structure where no force is acting?

For a thin circular curved member where the radius to the centroidal axis (R) is much larger than the section thickness (h), specifically R/h > 10, how can the strain energy (U) be approximated?

What is the classification for a column that is long and has a centrally applied compressive load?

What is the critical unit load (Pcr/A) for a column that obeys the Euler formula dependent on?

For a steel column, what is the limiting slenderness ratio (l/k)1, which separates long columns (Euler) from intermediate columns (Johnson), if the critical unit load at this transition is set to be half the yield strength (Pcr/A = Sy/2)?

What is the secant column formula used to analyze?

To differentiate between a 'secant column' and a 'strut' (short compression member), a limiting slenderness ratio is defined based on the bending deflection being limited to 1 percent of the eccentricity. What is the formula for this limiting ratio, (l/k)2?

What is the recommended end-condition constant (C) for a column with one end fixed and one end free?

A round steel column with a diameter of 20 mm is 375 mm long with rounded ends (C=1). Its material yield strength is 500 MPa and E is 207 GPa. The critical load is 88 kN. Is this column classified as an Euler (long) or Johnson (intermediate) column?

For a round column of diameter d, what is the formula to find its diameter based on the Euler formula for a given critical load Pcr, length l, modulus E, and end-condition constant C?

A cantilever beam has a translational stiffness of kt and a rotational stiffness of kr at its support. For an end load F applied at length l, what is the component of deflection (y2) caused by the rotation of the spring kr?

What is the effective diameter (de) for a nonrotating solid round bar of diameter (d) when using the critical volume theory for calculating the size factor kb?

For a simply supported beam of length l with a uniform load w over its entire span, what is the slope at the left support (x=0)?

What is the torsional spring rate (k) of a uniform solid round bar of diameter d, length l, and shear modulus G?

In the analysis of a statically indeterminate beam using Procedure 1, what is the first step?

A steel step shaft has second area moments of I1 for section AB and 2*I1 for section BC. It is loaded as a cantilever with force F at the end (point A). What is the deflection at point A?

For the same steel step shaft from the previous question (cantilever with force F at A, step at midpoint B), what is the deflection at the midpoint B?

What is the primary failure mode of a long, centrally loaded column when the applied force reaches the critical load, Pcr?

For a rectangular column of cross-section h x b (with h <= b) where the end conditions are the same for buckling in both directions, about which axis will buckling occur?

What is the primary reason that a column failure is considered more dangerous than a beam failure?

An indeterminate beam has three support reactions but only two available static equilibrium equations. How many additional deflection equations are required to solve for the unknown reactions?

What is the spring constant (k) of a uniform bar of length l, area A, and modulus of elasticity E, when subjected to an axially loaded force?

When using singularity functions to find the bending moment M in a beam, from the load intensity equation q, how many integrations are required?

For a simply supported beam of length l with a uniform load w, what is the maximum deflection?

What is the primary physical reason for including a size factor (kb) when calculating the endurance limit of a part in bending or torsion?

What is the recommended end-condition constant (C) for a column with both ends rounded (or pivoted)?

Which method for beam deflection analysis is described as being 'based on Castigliano's theorem' and 'quite powerful for problems not suitable for the methods mentioned earlier'?

An indeterminate beam is fixed at one end and has a simple support at the other. If the redundant reaction is chosen to be the reaction force R2 at the simple support, which two beam cases from a standard table would be superposed to solve for R2?

In the analysis of a column with eccentric loading using the secant formula, what is the term 'ec/k^2' called?

A round column 1.5 m long is to be designed for a critical load of 88 kN. It is fixed-fixed, so a recommended C value of 1.2 is used. The material has E = 207 GPa. It is determined that a cross section of h=23mm by b=92mm is needed. What is the slenderness ratio for this rectangular column?

For a column to be analyzed using the Euler formula, how must its actual slenderness ratio (l/k) relate to the limiting slenderness ratio (l/k)1?

What is the physical meaning of the spring constant, k?