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Deflection line of a cantilever beam

In this exercise the bending line for a cantilevered beam with two opposite loads is calculated according to the Bernoulli beam theory.

Task

A beam fixed on one side is loaded by the opposing forces F. The bending line for the beam has to be determined!

Beam clamped on one side with two opposing forces
Beam clamped on one side with two opposing forces

Solution

To determine the bending line, the internal forces of the beam must be determined. Only the bending moment is relevant for the bending, i.e. in the following equations only the moment equilibria are established. The beam is divided into two sections.

Two sections of the beam
Two sections of the beam

In this case, it is not necessary to determine the support reactions of the fixed restraint, so that the internal forces for section I can be started immediately.

Determination of the bending moments

Left-turning moments are positive in the following. The internal forces are applied negatively on a negative cutting edge.

Section I

Internal forces and moments in section I
Internal forces and moments in section I

The moment balance in section I is

\[ \tag{1} \sum M(x) = 0 = -M_{bI} + F \cdot (l-x) -- F \cdot (2 \cdot l -x) \]

\[ \tag{2} M_{bI} = -- F \cdot l \]

Section II

Internal forces and moments in section II
Internal forces and moments in section II

The moment balance in section II is

\[ \tag{3} \sum M(x) = 0 = -M_{bII} -- F \cdot (2 \cdot l -x) \]

\[ \tag{4} M_{bI} = F \cdot x -- 2 \cdot F \cdot l \]

Bending lines

The bending lines are determined on the basis of the Bernoulli beam bending. The basic relation is

\[ w” = \frac{-M_b}{E \cdot I} \]

Here, w” is the second derivative of the bending line, E is the Young’s modulus and I is the area moment of inertia. The bending line w is obtained by double integration. It follows from this for the present case

Section I

\[ \tag{5} E \cdot I \cdot w”_I = F \cdot l \]

\[ \tag{6} E \cdot I \cdot w’_I = F \cdot l \cdot x + c_1\]

\[ \tag{7} E \cdot I \cdot w_I = \frac{1}{2} F \cdot l \cdot x^2 + c_1 \cdot x + c_2 \]

Section II

\[ \tag{8} E \cdot I \cdot w”_{II} = -- F \cdot x + 2 \cdot F \cdot l \]

\[ \tag{9} E \cdot I \cdot w’_{II} = -- \frac{1}{2} F \cdot x^2 + 2 \cdot F \cdot l \cdot x + c_3 \]

\[ \tag{10} E \cdot I \cdot w_{II} = -- \frac{1}{6} F \cdot x^3 + F \cdot l \cdot x^2 + c_3 \cdot x + c_4 \]

In order to be able to determine the constants of integration c1 to c4, the boundary and transition conditions must be established.

Boundary and transition conditions

wI(x=0) = 0

The deflection at the point x = 0 is zero.

\[ \require{cancel} \]

\[ \tag{11} 0 = \frac{1}{E \cdot I} \left( \bcancel{\frac{1}{2}F \cdot l \cdot 0^2} + \bcancel{c_1 \cdot 0} + c_2 \right) \]

\[ \tag{12} c_2 = 0 \]

w’I(x=0) = 0

The angle at the point x = 0 is equal to zero.

\[ \require{cancel} \]

\[ \tag{13} 0 = \frac{1}{E \cdot I} \left(\bcancel{ F \cdot l \cdot 0} + c_1 \right) \]

\[ \tag{14} c_1 = 0 \]

wI(x=l) = wII(x=l)

The deflection of the two bending lines at the point x = l is equal.

\[ \require{cancel} \]

\[ \tag{15} \frac{1}{2} F \cdot l \cdot l^2 + \bcancel{c_1 \cdot l} + \bcancel{c_2} = -- \frac{1}{6} F \cdot l^3 + F \cdot l \cdot l^2 + c_3 \cdot l + c_4 \]

\[ \tag{16} \frac{1}{2} F \cdot l^3 = \frac{5}{6} F \cdot l^3 + c_3 \cdot l + c_4 \]

w’I(x=l) = w’II(x=l)

The angle of the two bending lines at the point x = l is the same.

\[ \require{cancel} \]

\[ \tag{17} F \cdot l \cdot l + \bcancel{c_1} = -- \frac{1}{2} F \cdot l^2 + 2 \cdot F \cdot l \cdot l + c_3 \]

From this context the integration constant c3 can be resolved.

\[ \tag{18} c_3 = -\frac{1}{2}F \cdot l^2 \]

And finally c4:

\[ \tag{19} \frac{1}{2} F \cdot l^3 = \frac{5}{6} F \cdot l^3 -- \frac{1}{2}F \cdot l^3 + c_4 \]

\[ \tag{20} c_4 = \frac{1}{6}F \cdot l^3 \]

The equations of the bending lines are thus

\[ \tag{21} w_I = \frac{1}{2 E \cdot I} F \cdot l \cdot x^2 \]

\[ \tag{22} w_{II} = \frac{F}{E \cdot I} \left( -- \frac{1}{6} x^3 + l \cdot x^2 -\frac{1}{2} l^2 \cdot x + \frac{1}{6} l^3 \right) \]