Determinate frames: axial force, shear and moment

A frame is a structure whose members are joined by at least some rigid connections, so that members carry bending moment, shear and axial force together. Portal frames for garages and warehouses, rigid bents in buildings of several stories, sign structures and crane brackets are all frames. Most real frames are statically indeterminate, but determinate frames are common enough in their own right (three-hinged portals, cantilever brackets, frames on a pin and a roller) and they are the building blocks for the indeterminate methods of later courses. The skills in this lesson, drawing free bodies of members and joints and plotting internal forces around corners, carry straight over.

From beams to frames

A frame member is analyzed exactly like a beam once you know the forces acting on its ends. The difference is that frames turn corners. At a rigid joint, the end moment of one member passes into the next, and the shear of one member becomes axial force in the other when they meet at a right angle. Tracking these transfers is the whole art. The procedure is reliable if you follow it step by step.

• Check stability and determinacy, and find the support reactions from equilibrium of the whole frame, plus any equations of condition at hinges.
• Cut each member free of its joints, and draw its free-body diagram with the end forces (axial, shear, moment) acting on it. Forces on the two sides of a joint are equal and opposite.
• Work from a support, where the end forces are known from the reactions, around the frame member by member. At each rigid joint, use joint equilibrium to pass the moment, and to convert the shear and axial force of one member into the axial force and shear of the next.
• Draw the axial, shear and moment diagrams for each member along its own axis, and check that every joint is in equilibrium.

Sign conventions for frames

Because members point in different directions, a positive sign for shear or moment depends on which side you look from. Two practical conventions avoid the confusion. For axial force, tension is positive, which needs no reference direction. For moment, structural engineers usually draw the moment diagram on the tension side of each member, and label it with a magnitude only. A moment diagram plotted on the outside of a column at the knee of a portal means tension on the outside face there, which immediately tells a concrete designer where the reinforcement goes and a steel designer which flange is in compression. For shear, pick a direction along each member (for example, viewing each column from the inside of the frame) and stick to it.

Note what the roller at D does: it makes the frame determinate but sends all the horizontal load to column AB, which must be designed for the bending that results. Real portals usually have pinned or fixed bases on both sides, which shares the horizontal load and makes the frame indeterminate. You will analyze those in CIV-321, but the free-body habits are the same.

Joint equilibrium at a corner

At a rigid corner joint where a column meets a beam at 90°, three equations hold. Horizontal equilibrium relates the column's shear to the beam's axial force (plus any horizontal load at the joint). Vertical equilibrium relates the column's axial force to the beam's shear (plus any vertical load at the joint). Moment equilibrium requires the end moments of the two members to be equal and opposite on the joint, so the moment diagram is continuous around the corner, staying on the same side (inside or outside) of the frame. If three or more members meet, the moments of all members at the joint must sum to any applied joint moment, which is zero in most cases.

A common slip is to forget that the moment continues around the corner. If the column has tension on its inside face at the top, the beam has tension on its bottom (inside) face at the same end. Plotting the diagrams on the tension side makes this visible: the curve wraps around the inside of the corner.

Three-hinged portal frames

In lesson 3 we found the reactions of the salt shed's three-hinged frame. The inward thrust at the base produces large moments at the knees, and these govern the design. The hinge at the crown forces the moment to zero there, so the rafters are in mostly negative moment (tension on the outside), unlike a simple beam.

Axial force and shear in inclined members

For an inclined member such as the rafter, resolve the forces at a section into components along and perpendicular to the member. If the rafter makes an angle θ with the horizontal, and the free body to one side of the section has a net vertical force V (up) and net horizontal force H (inward), then the axial force (compression positive here) is \(N = H\cos\theta + V\sin\theta\) and the shear perpendicular to the member is \(Q = V\cos\theta - H\sin\theta\). For the salt shed rafter just right of the knee (\(\theta = 21.8°\)), H is the 12.0-kip thrust and V is the 25-kip vertical reaction, giving an axial compression of about 20.4 kips and a shear of about 18.7 kips. The rafter is therefore a beam-column, designed for combined compression and bending.

Cantilever frames and brackets

Many frames are cantilevers fixed at one support: a sign structure, a canopy bracket, a crane runway bracket or a pipe support. Their analysis needs no reactions at all. Start at the free end and work toward the fixed support, cutting the frame at each section and summing the forces and moments on the free part. The crane bracket on a garage column, for example, carries the runway beam's reaction at an eccentricity from the column centerline, so it delivers to the column a vertical force and a moment equal to that force times the eccentricity. That moment then travels down the column to its base.

Frame moments by computer

Sofia checks the three-hinged frame by computing the moment at many points around the frame from the reactions. The same short function works for any determinate plane frame once the reactions are known, because the moment at a section is just the sum of the moments of the forces on one side of it.

# Three-hinged frame: bases A(0,0), E(50,0); knees B(0,16), D(50,16); crown C(25,26); load 1.0 kip/ft on plan
w, span, h, rise = 1.0, 50.0, 16.0, 10.0
Ay = w * span / 2
Ax = (Ay * span / 2 - w * (span / 2) ** 2 / 2) / (h + rise)      # crown hinge condition, inward
print(f"Ay = {Ay:.1f} kips, thrust H = {Ax:.2f} kips")

def M_left(x, y):
    """Moment at section (x, y) on the left half; sagging (tension inside) positive."""
    return Ay * x - Ax * y - w * x ** 2 / 2

print("Column AB:")
for y in [0, 4, 8, 12, 16]:
    print(f"  y = {y:4.1f} ft   M = {M_left(0.0, y):7.1f} kip*ft")
print("Rafter BC:")
for i in range(6):
    x = i * 5.0; y = h + rise * x / (span / 2)
    print(f"  x = {x:4.1f} ft   M = {M_left(x, y):7.1f} kip*ft")

Ay = 25.0 kips, thrust H = 12.02 kips
Column AB:
  y =  0.0 ft   M =     0.0 kip*ft
  y =  4.0 ft   M =   -48.1 kip*ft
  y =  8.0 ft   M =   -96.2 kip*ft
  y = 12.0 ft   M =  -144.2 kip*ft
  y = 16.0 ft   M =  -192.3 kip*ft
Rafter BC:
  x =  0.0 ft   M =  -192.3 kip*ft
  x =  5.0 ft   M =  -103.8 kip*ft
  x = 10.0 ft   M =   -40.4 kip*ft
  x = 15.0 ft   M =    -1.9 kip*ft
  x = 20.0 ft   M =    11.5 kip*ft
  x = 25.0 ft   M =     0.0 kip*ft

The table confirms the shape: the moment grows linearly up the column to its peak at the knee, then decreases along the rafter to zero at the crown hinge, with only a small reversal just before the crown.