In the conjugate-beam method, if the M/EI diagram on the real beam is negative, what is the direction of the distributed load on the conjugate beam?

Correct answer: Downward, away from the beam.

Explanation

The direction of the load on the conjugate beam is a critical convention. A positive M/EI diagram (which causes a beam to bend concave up) is treated as an upward, or lifting, load on the conjugate beam.

Other questions

Question 1

What does the elastic curve of a beam or frame represent?

Question 2

According to the sign convention discussed in Chapter 8, a positive internal moment tends to bend a horizontal beam member in which direction?

Question 3

What does a point of inflection on the elastic curve of a beam correspond to?

Question 4

What is the primary reason for placing reinforcing steel rods in concrete beams in regions of tension?

Question 5

In the elastic-beam theory, what is the relationship `1/ρ = M/EI` called?

Question 6

The simplified differential equation for the elastic curve is `d²v/dx² = M/EI`. What key assumption, known as small deflection theory, is made to arrive at this simplified form?

Question 7

When using the double integration method, what is the purpose of boundary and continuity conditions?

Question 8

A simply supported beam of length 10 m is subjected to a uniform design loading of 4 kN/m. The support reactions are 20 kN at each end. What is the equation for the internal moment M(x) as a function of x, where x is measured from the left support?

Question 9

A cantilevered beam of length L is subjected to a couple moment M0 at its free end. What is the maximum slope of the beam in terms of M0, L, and EI?

Question 10

What does the first moment-area theorem allow you to calculate?

Question 11

What does the second moment-area theorem state regarding the vertical deviation of tangents?

Question 12

A cantilever beam is fixed at end A and is 30 ft long. A 2-kip load at the free end C creates a triangular M/EI diagram that is 0 at C and -60/EI at A. Using the first moment-area theorem, what is the slope at point C, θC?

Question 13

For a cantilever beam fixed at A with length L, a downward point load P at the free end B creates a triangular M/EI diagram. Using the second moment-area theorem, what is the tangential deviation of B with respect to A, t_B/A?

Question 14

In the conjugate-beam method, what principle from statics is used to find the slope of the real beam?

Question 15

If a real beam has an internal hinge, what is the corresponding feature on the conjugate beam?

Question 16

How is the conjugate beam loaded?

Question 17

A simply supported beam of length 12 m has a moment diagram that results in a positive (upward) parabolic load on its conjugate beam. The total magnitude of this parabolic load is 9000/EI, and its centroid is 6 m from the left end. What is the slope at the left support of the real beam?

Question 18

A simply supported beam has a pin support at A and a roller at B. For analysis using the conjugate-beam method, what are the supports of the conjugate beam at A' and B'?

Question 19

A steel beam (E = 200 GPa, I = 60(10^6) mm^4) is subjected to loads that result in a required slope calculation of θ = -112.5 kN-m^2 / EI. What is the final calculated slope in radians?

Question 20

Which of the following deflection methods is based on a semigraphical technique where the M/EI diagram is used to find the area and moment of the area?

Question 21

In the context of the moment-area theorems, what does the notation t_A/B represent?

Question 22

If the M/EI diagram between points A and B on a beam is positive, what does this imply about the change in slope, θ_B/A, according to the first moment-area theorem?

Question 23

What physical property does the product EI represent for a beam?

Question 24

A beam has an internal hinge at point B. When drawing the deflected shape (elastic curve), what is a key characteristic at point B?

Question 25

A cantilever beam fixed at A and free at C is 7 meters long. A couple moment of 500 N-m is applied at C. The beam's EI_BC is half of EI_AB, where B is 4 meters from A. The M/EI diagram relative to EI_BC has a constant value of 250/EI_BC from A to B, and 500/EI_BC from B to C. Using the second moment-area theorem, what is the deflection at C?

Question 26

For the symmetric, simply supported beam in Figure 8-18a, with a downward load at the center, the tangent to the elastic curve at the center point D is horizontal. How can the slope at the support C, θ_C, be found using moment-area theorems?

Question 27

Which of the deflection methods covered in Chapter 8 is described as relying 'only on the principles of statics, and hence its application will be more familiar'?

Question 28

In the double integration method for a beam with discontinuous loads, multiple x coordinates are used for different regions. What condition must be satisfied at the point where two regions meet?

Question 29

A steel beam (E=29(10^3) ksi, I=800 in^4) has a downward triangular M/EI diagram on its conjugate beam. The area of the diagram is 562.5 k-ft^2/EI and the total length is 30 ft. The beam is a cantilever fixed at A (real). What is the slope at the free end B (real)?

Question 30

When is it particularly advantageous to use the moment-area theorems?

Question 31

What type of support on a real beam restricts both displacement and rotation?

Question 32

A cantilever beam fixed at its left end has a length of 10 m and is subjected to a uniform downward load of 12 kN/m. What is the value of the internal moment M at the fixed support?

Question 34

What does a zero value for the M/EI diagram at a point signify about the elastic curve?

Question 35

For the compound beam in Figure 8-14a, which has an internal roller at B, why are two x-coordinates required for analysis using the double integration method?

Question 36

A simply supported beam AB of length L is subjected to a point load P at its center. What is the maximum deflection of the beam in terms of P, L, and EI?

Question 37

What is a key difference in applicability between the method of virtual work and Castigliano's theorem for finding beam deflections?

Question 38

In the conjugate-beam method, what does the shear at the free end of a conjugate beam correspond to?

Question 39

Why must deflections of structures be limited in design?

Question 40

What are the units of the angular flexibility coefficient, α, as used in the analysis of indeterminate beams?

Question 41

A beam's M/EI diagram is a rectangle with height 10/EI and length 5m. What is the change in slope between the two ends of this 5m segment?

Question 42

What is the primary cause of deflection in trusses, as compared to beams and frames?

Question 43

A fixed-connected joint in a frame deflects under load. What is the relationship between the rotations of the members connected at that joint?

Question 44

In the context of the conjugate-beam method, a statically determinate real beam will have a conjugate beam that is:

Question 45

A symmetric beam is symmetrically loaded, causing the M/EI diagram to also be symmetric. Where does the maximum deflection occur?

Question 46

What is the primary advantage of the tabular method for integrating the product ∫mM dx, as mentioned in Section 9.7?

Question 47

If the calculated deflection `v` using the double integration method is a positive value, what does this signify about the direction of displacement?

Question 48

A simply supported beam of length 6m has a concentrated load of 8 kN at its center (3m from the left support). What is the moment M(x) for the left half of the beam (0 < x < 3), where x is measured from the left support?

Question 49

What does a negative sign for a calculated tangential deviation, such as t_A/B, indicate?

Question 50

A cantilever beam of length L is subjected to a uniform downward load w. The M(x) function, with x from the free end, is M(x) = -wx^2/2. Using the conjugate-beam method, what is the deflection at the free end?