Question 9.3: A simple beam AB supports a concentrated load P acting at di...
A simple beam AB supports a concentrated load P acting at distances a and b from the left-hand and right-hand supports, respectively (Fig. 9-12a).
Determine the equations of the deflection curve, the angles of rotation θ_{A} and θ_{B} at the supports, the maximum deflection δ_{max}, and the deflection δ_{C} at the midpoint C of the beam (Fig. 9-12b). Note: The beam has length L and constant flexural rigidity EI.
Use a four-step problem-solving approach.
1. Conceptualize: Internal shear V(x) and moment M(x) are discontinuous at load point P. Two free-body diagrams are required to obtain moment expressions over the entire length of the beam (see Figs. 9-13a and b).
Bending moments in the beam: In this example, the bending moments are expressed by two equations—one for each part of the beam. Use the free-body diagrams of Fig. 9-13 to arrive at
M=\frac{P b x}{L} (0 \leq x \leq a) (9-33a)
M=\frac{P b x}{L}-P(x-a) \quad(a \leq x \leq L) (9-33b)
2. Categorize:
Differential equations of the deflection curve: The differential equations for the two parts of the beam are obtained by substituting the bending-moment expressions [Eqs. (9-33a and b)] into Eq. (9-16a). The results are
E I v^{\prime \prime}=\frac{P b x}{L} \quad(0 \leq x \leq a) (9-34a)
E I v^{\prime \prime}=\frac{P b x}{L}-P(x-a) \quad(a \leq x \leq L) (9-34b)
E I v^{\prime \prime}=M (9-16)
3. Analyze:
Slopes and deflections of the beam: The first integrations of the two differential equations yield the following expressions for the slopes:
E I v^{\prime}=\frac{P b x^{2}}{2 L}+C_{1} (0 \leq x \leq a) (a)
E I v^{\prime}=\frac{P b x^{2}}{2 L}-\frac{P(x-a)^{2}}{2}+C_{2} \quad(a \leq x \leq L) (b)
in which C_{1} and C_{2} are constants of integration. A second pair of integrations gives the deflections:
E I v=\frac{P b x^{3}}{6 L}+C_{1} x+C_{3} (0 \leq x \leq a) (c)
E I v=\frac{P b x^{3}}{6 L}-\frac{P(x-a)^{3}}{6}+C_{2} x+C_{4} \quad(a \leq x \leq L) (d)
These equations contain two additional constants of integration, making a total of four constants to be evaluated.
Constants of integration: The four constants of integration can be found from the following four conditions:
i. At x = a, the slopes v′ for the two parts of the beam are the same.
ii. At x = a, the deflections v for the two parts of the beam are the same.
iii. At x = 0, the deflection v is zero.
iv. At x = L, the deflection v is zero.
The first two conditions are continuity conditions based upon the fact that the axis of the beam is a continuous curve. Conditions (iii) and (iv) are boundary conditions that must be satisfied at the supports.
Condition (i) means that the slopes determined from Eqs. (a) and (b) must be equal when x = a; therefore,
Condition (ii) means that the deflections found from Eqs. (c) and (d) must be equal when x = a; therefore,
\frac{P b a^{3}}{6 L}+C_{1} a+C_{3}=\frac{P b a^{3}}{6 L}+C_{2} a+C_{4}Inasmuch as C_{1} = C_{2}, this equation gives C_{3} = C_{4}.
Next, apply condition (iii) to Eq. (c) and obtain C_{3} = 0; therefore,
C_{3}=C_{4}=0 (e)
Finally, apply condition (iv) to Eq. (d) and obtain
\frac{P b L^{2}}{6}-\frac{P b^{3}}{6}+C_{2} L=0Therefore,
C_{1}=C_{2}=-\frac{P b\left(L^{2}-b^{2}\right)}{6 L} (f)
Equations of the deflection curve: Now substitute the constants of integration [Eqs. (e) and (f)] into the equations for the deflections [Eqs. (c) and (d)] and obtain the deflection equations for the two parts of the beam. The resulting equations, after a slight rearrangement, are
v=-\frac{P b x}{6 L E I}\left(L^{2}-b^{2}-x^{2}\right) (0 \leq x \leq a) (9-35a)
v=-\frac{P b x}{6 L E I}\left(L^{2}-b^{2}-x^{2}\right)-\frac{P(x-a)^{3}}{6 E I} (a \leq x \leq L) (9-35b)
The first of these equations gives the deflection curve for the part of the beam to the left of the load P, and the second gives the deflection curve for the part of the beam to the right of the load.
The slopes for the two parts of the beam can be found either by substituting the values of C_{1} and C_{2} into Eqs. (a) and (b) or by taking the first derivatives of the deflection equations [Eqs. (9-35a and b)]. The resulting equations are
v^{\prime}=-\frac{P b}{6 L E I}\left(L^{2}-b^{2}-3 x^{2}\right) (0 \leq x \leq a) (9-36a)
v^{\prime}=-\frac{P b}{6 L E I}\left(L^{2}-b^{2}-3 x^{2}\right)-\frac{P(x-a)^{2}}{2 E I} \quad(a \leq x \leq L) (9-36b)
The deflection and slope at any point along the axis of the beam can be calculated from Eqs. (9-35) and (9-36).
Angles of rotation at the supports: To obtain the angles of rotation \theta_{A} and \theta_{B} at the ends of the beam (Fig. 9-12b), substitute x = 0 into Eq. (9-36a) and x = L into Eq. (9-36b):
\theta_{A}=-v^{\prime}(0)=\frac{P b\left(L^{2}-b^{2}\right)}{6 L E I}=\frac{P a b(L+b)}{6 L E I} (9-37a)
\theta_{B}=v^{\prime}(L)=\frac{P b\left(2 L^{2}-3 b L+b^{2}\right)}{6 L E I}=\frac{P a b(L+a)}{6 L E I} (9-37b)
Note that the angle \theta_{A} is clockwise and the angle \theta_{B} is counter-clockwise, as shown in Fig. 9-12b.
The angles of rotation are functions of the position of the load and reach their largest values when the load is located near the midpoint of the beam. In the case of the angle of rotation \theta_{A}, the maximum value of the angle is
\left(\theta_{A}\right)_{\max }=\frac{P L^{2} \sqrt{3}}{27 E I} (9-38)
and occurs when b = L / \sqrt{3} = 0.577 L (or a = 0.423 L). This value of b is obtained by taking the derivative of \theta_{A} with respect to b [using the first of the two expressions for \theta_{A} in Eq. (9-37a)] and then setting it equal to zero.
Maximum deflection of the beam: The maximum deflection \delta_{max} occurs at point D (Fig. 9-12b) where the deflection curve has a horizontal tangent. If the load is to the right of the midpoint, that is, if a > b, point D is in the part of the beam to the left of the load. Locate this point by equating the slope v′ from Eq. (9-36a) to zero and solving for the distance x, which is now denoted as x_{1}. In this manner, obtain the following formula for x_{1}:
x_{1}=\sqrt{\frac{L^{2}-b^{2}}{3}} \quad(a \geq b) (9-39)
From this equation, note that as the load P moves from the middle of the beam (b = L / 2) to the right-hand end (b = 0), the distance x_{1} varies from L/2 to L /\sqrt{3} = 0.577 L. Thus, the maximum deflection occurs at a point very close to the midpoint of the beam, and this point is always between the midpoint of the beam and the load.
The maximum deflection \delta_{max} is found by substituting x_{1} [from Eq. (9-39)] into the deflection equation [Eq. (9-35a)] and then inserting a minus sign:
\delta_{\max }=-(v)_{x=x 1}=\frac{P b\left(L^{2}-b^{2}\right)^{3 / 2}}{9 \sqrt{3} L E I} \quad(a \geq b) (9-40)
The minus sign is needed because the maximum deflection is downward (Fig. 9-12b), whereas the deflection v is positive upward.
The maximum deflection of the beam depends on the position of the load P, that is, on the distance b. The maximum value of the maximum deflection (the “max-max” deflection) occurs when b = L / 2 and the load is at the midpoint of the beam. This maximum deflection is equal to PL³ / 48EI .
Deflection at the midpoint of the beam: The deflection \delta_{C} at the midpoint C when the load is acting to the right of the midpoint (Fig. 9-12b) is obtained by substituting x = L / 2 into Eq. (9-35a), as
\delta_{C}=-v\left(\frac{L}{2}\right)=\frac{P b\left(3 L^{2}-4 b^{2}\right)}{48 E I} \quad(a \geq b) (9-41)
4. Finalize: Because the maximum deflection always occurs near the midpoint of the beam, Eq. (9-41) yields a close approximation to the maximum deflection. In the most unfavorable case (when b approaches zero), the difference between the maximum deflection and the deflection at the midpoint is less than 3% of the maximum deflection, as demonstrated in Problem 9.3-9.
Special case (load at the midpoint of the beam): An important special case occurs when the load P acts at the midpoint of the beam (a = b = L / 2). Then the following results are obtained from Eqs. (9-36a), (9-35a), (9-37), and (9-40), respectively:
v^{\prime}=-\frac{P}{16 E I}\left(L^{2}-4 x^{2}\right) \quad\left(0 \leq x \leq \frac{L}{2}\right) (9-42)
v=-\frac{P x}{48 E I}\left(3 L^{2}-4 x^{2}\right) \quad\left(0 \leq x \leq \frac{L}{2}\right) (9-43)
\theta_{A}=\theta_{B}=\frac{P L^{2}}{16 E I} (9-44)
\delta_{\max }=\delta_{C}=\frac{P L^{3}}{48 E I} (9-45)
Since the deflection curve is symmetric about the midpoint of the beam, the equations for v′ and v are given only for the left-hand half of the beam in Eqs. (9-42) and (9-43). If needed, the equations for the right-hand half can be obtained from Eqs. (9-36b) and (9-35b) by substituting a = b = L / 2.
