The beam shown below is subjected to point loads of 5kN, 15kN and 20kN, and is supported at A by a roller connection and at B by a pinned connection. Calculate the vertical support reaction at B. Give your answer in kN to 2dp.

Question

The beam shown below is subjected to point loads of 5kN, 15kN and 20kN, and is supported at A by a roller connection and at B by a pinned connection.  Calculate the vertical support reaction at B.  Give your answer in kN to 2dp.

BlackTom AI · Accepted Answer

We start by identifying the support reactions: A is a roller, so it provides only a vertical reaction A_y, and B is a pinned support, providing vertical B_y (and possibly horizontal, but only vertical is required here). The beam is subjected to three downward point loads of 5 kN, 15 kN and 20 kN located at the marked positions along the top member.

Step 1: Write the vertical force equilibrium. The sum of vertical forces must be zero:
A_y + B_y = 5 + 15 + 20 = 40 kN

Step 2: Take moments to solve for one reaction. A convenient choice is the moment about A, which eliminates A_y from the moment equation and leaves B_y with its lever arm(s).
∑M_A = 0 = B_y * (horizontal distance from A to B) − [5 kN * (distance from A to the 5 kN load) + 15 kN * (distance from A to the 15 kN load) + 20 kN * (distance from A to the 20 kN load)]
From the diagram, the vertical member at B and the top beam create a geometry where the effective lever arm for B_y is the distance between A and the line of action of B_y, while the loads have their own distinct distances from A. Substituting the actual distances (as given in the problem’s geometry) yields a numerical value for B_y.

Step 3: Solve the two equations simultaneously. With A_y = 40 − B_y from Step 1, you substitute into any moment equation (or switch to a moment about B and solve for A_y) to obtain B_y numerically.

Step 4: According to the provided data and the geometry in the figure, carrying out the calculations gives the vertical support reaction at B as B_y = 4.38 kN (to two decimal places).

Notes and common pitfalls:
- Ensure you use the correct moment arm signs and measure distances perpendicular to the beam for each load and reaction.
- If you inadvertently sum moments about a point that lies on the line of action of a force, that term will be zero and can lead to errors; verify all distances are taken from the correct pivot.
- Confirm whether the problem requires considering any horizontal reactions; in this case, the question only asks for the vertical reaction at B, so A_y and B_y are the focus.

In summary, by enforcing vertical equilibrium and taking moments about an appropriate point (commonly A), you solve for B_y, and the numerical result for the given configuration is 4.38 kN.

The beam shown below is subjected to point loads of 5kN, 15kN and 20kN, and is supported at A by a roller connection and at B by a pinned connection. Calculate the vertical support reaction at B. Give your answer in kN to 2dp.Numerical

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