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002714

2023-04-26

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Internal Forces

In the navigator, define the internal forces and moments to be displayed on the surfaces. The table lists the internal forces and moments of each surface according to the specifications set in the Result Table Manager .

Selecting Surface Internal Forces in Navigator

Basic Internal Forces of Surfaces in Table

The surface internal forces are subdivided into three categories:

Basic Internal Forces: internal forces and moments in the direction of the surface axes
Principal Internal Forces: internal forces and moments in the direction of the principal axes
Design Internal Forces: internal forces and moments according to DIN V ENV 1992-1-1 [1] Appendix 2, A 2.8 and A 2.9

Basic Internal Forces

In contrast to member internal forces, surface internal forces are symbolized by small letters. The integral definition of the bending moments m_x and m_y shows that moments are related to the directions of the surface axes where the corresponding normal stresses are created.

When curved surfaces are analyzed, internal forces and moments refer to the local axes of the finite elements. You can set the display of the "FE Axis Systems" in the navigator.

Displaying FE Axis Systems

Important

There is a fundamental difference in the understanding of surface internal forces and member internal forces: while the member moment M_y "rotates" about the local member axis y, the surface moment m_y acts in the direction of the local surface axis y; that is, about the axis x of this surface.

The following image shows the basic internal forces and stresses of a surface symbolically.

Surface Internal Forces and Surface Stresses

The surface moments as well as the shear stresses that are perpendicular to the surface show a parabolic distribution along the surface thickness.

Sign

The algebraic signs indicate the side of the surface on which the internal force or moment is available: If the global Z-axis is directed downwards, positive internal forces and moments generate tensile stresses on the positive side of the surface (that is, in the direction of the positive surface axis z). Negative internal forces and moments result in compressive stresses on the positive side of the surface. If the global Z-axis is aligned with upwards, the signs are reversed accordingly.

The basic internal forces and moments for a downward Z-axis are determined with the following equations.

m_{x} = \int_{- \frac{d}{2}}^{+ \frac{d}{2}} σ_{x} z d z

Bending Moment m_x (Creating Stresses in Direction of Local x-Axis)

m_{y} = \int_{- \frac{d}{2}}^{+ \frac{d}{2}} σ_{y} z d z

Bending Moment m_y (Creating Stresses in Direction of Local y-Axis)

m_{x y} = m_{y x} = \int_{- \frac{d}{2}}^{+ \frac{d}{2}} τ_{x y} z d z

Torsional Moments m_xy and m_yx

v_{x} = \int_{- \frac{d}{2}}^{+ \frac{d}{2}} τ_{x z} d z

Shear Force v_x

v_{y} = \int_{- \frac{d}{2}}^{+ \frac{d}{2}} τ_{y z} d z

Shear Force v_y

n_{x} = \int_{- \frac{d}{2}}^{+ \frac{d}{2}} σ_{x} d z

Axial Force n_x in Direction of Local x-Axis

n_{y} = \int_{- \frac{d}{2}}^{+ \frac{d}{2}} σ_{y} d z

Axial Force n_y in Direction of Local y-Axis

n_{x y} = \int_{- \frac{d}{2}}^{+ \frac{d}{2}} τ_{x y} d z

Shear Flow n_xy

Principal Internal Forces

While the basic internal forces and moments refer to the more or less freely created xyz-coordinate system of a surface, the principal internal forces and moments represent the extreme values of the internal forces and moments within a surface element. To determine the principal internal forces, the basic internal forces are transformed in the directions of both principal axes. The principal axes 1 (maximum value) and 2 (minimum value) are arranged orthogonally.

The principal internal forces and moments are determined from the basic internal forces and moments as follows:

m_{1} = \frac{1}{2} [m_{x} + m_{y} + \sqrt{{(m_{x} - m_{y})}^{2} + 4 {m_{x y}}^{2}}]

Bending Moment m₁ in Direction of Principal Axis 1

m_{2} = \frac{1}{2} [m_{x} + m_{y} - \sqrt{{(m_{x} - m_{y})}^{2} + 4 {m_{x y}}^{2}}]

Bending Moment m₂ in Direction of Principal Axis 2

α_{b} = \frac{1}{2} [\arctan (\frac{2 m_{x y}}{m_{x} - m_{y}})]

Angle α_b Between Local Axis x (or y) and Principal Axis 1 (or 2)

m_{T, \max, b} = \frac{\sqrt{{(m_{x} - m_{y})}^{2} + 4 {m_{x y}}^{2}}}{2}

Maximum Torsional Moment m_T,max,b

v_{\max, b} = \sqrt{{v_{x}}^{2} + {v_{y}}^{2}}

Maximum Resulting Shear Force v_max,b from Bending Components

β_{b} = \arctan \frac{v_{y}}{v_{x}}

Angle β_b Between Principal Shear Force v-max,b and Local Axis x

n_{1} = \frac{1}{2} [n_{x} + n_{y} + \sqrt{{(n_{x} - n_{y})}^{2} + 4 {n_{x y}}^{2}}]

Axial Force n₁ in Direction of Principal Axis 1

n_{2} = \frac{1}{2} [n_{x} + n_{y} - \sqrt{{(n_{x} - n_{y})}^{2} + 4 {n_{x y}}^{2}}]

Axial Force n₂ in Direction of Principal Axis 2

α_{m} = \frac{1}{2} [\arctan (\frac{2 n_{x y}}{n_{x} - n_{y}})]

Angle α_m Between Axis x and Principal Axis 1 (for Axial Force n₁)

v_{\max, m} = \frac{\sqrt{{(n_{x} - n_{y})}^{2} + 4 {n_{x y}}^{2}}}{2}

Maximum Shear Force v_max,m from Membrane Components

You can graphically display the principal axes directions α_b (for bending moments), β_b (for shear forces), and α_m (for axial forces) as trajectories.

Displaying Trajectories of Principal Axes

For example, if you display the angle α_b, you can also see the magnitudes of the respective principal moments. The trajectories are scaled to the values of the moments m₁ and m₂.

Design Internal Forces

The design moments and axial forces are based on the approach described in DIN V ENV 1992‑1‑1 [1], Appendix 2, A 2.8 and A 2.9. This may be a help when performing reinforced concrete design checks manually. The design internal forces are irrelevant for the "Concrete Design" add-on, as the Baumann method [2] is used there.

Info

The design moments and axial forces must not be combined! As explained in [1], Appendix 2.8, the moments refer exclusively to slab reinforcements. Axial forces are to be used for designing diaphragm elements according to [1], Appendix 2.9.

The design internal forces are specified in Chapter 8.17 of the RFEM 5 manual.

References

DIN V ENV 1992-1-1. (1992). Planung von Stahlbeton- und Spannbetontragwerken – Teil 1: Grundlagen und Anwendungsregeln für den Hochbau. Beuth Verlag GmbH, Berlin.
Baumann, Germany. Th. Frage der Netzbewehrung von Flächentragwerken. Th. Bauingenieur, 47, 1972.

Parent Chapter

Results by Surface