Shear force bending moment

Considering a section at a distance $x$ from A:

$$V(x) = R_A$$ $$V(x) = 5$$

The shear force is constant in this section.

Considering a section at a distance $x$ from A:

$$V(x) = R_A - P$$ $$V(x) = 5 - 10$$ $$V(x) = -5$$

The shear force is constant in this section.

Bending Moment Calculation

Bending moment ($M(x)$) is the algebraic sum of the moments of all forces to the left of the section about that section. Counter-clockwise moments are positive, clockwise moments are negative.

Considering a section at a distance $x$ from A:

$$M(x) = R_A \times x$$ $$M(x) = 5x$$

At $x=0$ (support A), $M(0) = 5 \times 0 = 0$.
At $x=3$ (at the point load), $M(3) = 5 \times 3 = 15$.

Considering a section at a distance $x$ from A:

$$M(x) = R_A \times x - P \times (x - 3)$$ $$M(x) = 5x - 10(x - 3)$$ $$M(x) = 5x - 10x + 30$$ $$M(x) = -5x + 30$$

At $x=3$ (at the point load), $M(3) = -5 \times 3 + 30 = -15 + 30 = 15$.
At $x=6$ (support B), $M(6) = -5 \times 6 + 30 = -30 + 30 = 0$.

Summary of Shear Force and Bending Moment Functions

The shear force and bending moment functions for the beam are:

Shear Force ($V(x)$):

$$V(x) = \begin{cases} 5 & \text{for } 0 \le x < 3 \\ -5 & \text{for } 3 < x \le 6 \end{cases}$$

Bending Moment ($M(x)$):

$$M(x) = \begin{cases} 5x & \text{for } 0 \le x < 3 \\ -5x + 30 & \text{for } 3 < x \le 6 \end{cases}$$

The maximum bending moment occurs at the point load ($x=3$), with a value of 15 units.

The final answer is $\boxed{\text{Shear Force: } V(x) = \begin{cases} 5 & \text{for } 0 \le x < 3 \\ -5 & \text{for } 3 < x \le 6 \end{cases}, \text{Bending Moment: } M(x) = \begin{cases} 5x & \text{for } 0 \le x < 3 \\ -5x + 30 & \text{for } 3 < x \le 6 \end{cases}}$.